After a while it begins to feel relentless. Here come the national news magazines, and the suburban dailies, and the eastern papers, and Cable News Network and the "CBS Morning News"; here come Brussels and Tokyo television men, wondering if she might spare them a moment or two. Here comes the AM radio man, following her even into a late-night television appearance, asking about it again. "Aw, I've gone into that," the mayor says. "Enough's enough."

She grins, shrugs the radio man away, turns to see Dan Rather on the television pressing Walter Mondale for details about his running mate.

"Well," Rather says huffily, "if you won't tell us her name--"

And Dianne Feinstein starts laughing so hard she nearly doubles over.

Last Saturday, in a two-hour visit at his home in North Oaks, Minn., Walter Mondale interviewed Dianne Feinstein about her possibilities as Democratic candidate for the vice presidency of the United States. "A symbol of the very best in America," Mondale said afterward, as the two of them smiled broadly for reporters. It was a laboriously publicized visit, like nearly everything connected with the present fuss over the vice presidency; it has been duly noted that in his first 10 days of interviewing potential running mates, Mondale interviewed and thus publicly flattered Texas Sen. Lloyd Bentsen, a southerner; Los Angeles Mayor Tom Bradley, a black; and Dianne Feinstein, a woman.

A Jewish woman, at that.

A Jewish woman from San Francisco, which in other parts of the country has what advertising people might call an "image problem."

She knows the odds on this.

"My youngest stepdaughter, Eileen, spent the night with me last night, and we were all sitting in bed," Feinstein says. "And she said to me, 'Do you think anything'll happen?' And I said, 'No. But it's really an honor to be on that short list.' I'm one of a very few number of women, and I'm proud to be there. I mean, there's a certain sense of pride."

A kind of bruised sound is creeping into her voice.

"I mean, after all the b.s. you can take in this job, well, I can hold my head high," Feinstein says. "And that's nice, too."

The mayor of San Francisco is sitting, as she says this, on a bench alongside Stow Lake, which is a small and locally cherished man-made lake in Golden Gate Park. She is wearing a slightly sweaty T-shirt, blue warm-up pants, running shoes and tortoise-shell sunglasses. She is the mayor of an odd, fickle, 700,000-person city, a woman shoved into office by a double murder, elected to a standard term, subjected to an unyieldingly nasty recall campaign, upheld by a vast majority of the voters and ushered nearly without opposition into a second mayoral term.

She is tall, broad-shouldered, an athlete softened by the years, and she has nice crinkles around her eyes. People think she has no sense of humor. She has admitted to watching "Dynasty" when she is tired. She is coolly moderate in her political leanings, exasperating both left and right. Her husband, whom she married four years ago after she was widowed in 1978, climbs mountains and runs great distances. She has a deaf cat, likes Chinese porcelain, is said to be a mediocre cook, celebrated her 51st birthday last week, and is partial to flowers, particularly the small bright impatiens, which grows profusely in her garden.

By mid-July, when 25,000 people arrive in San Francisco for the Democratic National Convention, she will arguably have become the most celebrated mayor in the United States.

Bob Strauss, the former Democratic National Committee chairman and semiofficial party seer, calls her "one of the bright, gifted political leaders in either political party," but observes that she has little of the real national experience that a vice presidential candidate might need.

Douglas Schoen, whose private polling service has been monitoring public response to vice presidential candidates, found her the most popular female choice in the country four months ago, but says she recently dropped to second, one point behind New York Rep. Geraldine Ferraro.

It is hard to find anybody in San Francisco who thinks she will really be the candidate.

"It's not really fun," she says. She means the constant volleys of questions about the vice presidency, but there is also the convention besieging her: What about the convention, Mayor Feinstein, how about those cost overruns, will the police beat up the demonstrators, will the demonstrators block traffic, will anybody be able to get a restaurant reservation, Mayor Feinstein? Do you want a Cabinet post? Do you want to be president?

"I do think it does this," Feinstein says. "I spend a lot of time putting programs together and lobbying for money, and this is helpful in that direction. It's helpful with the state legislature. It's helpful with the Congress. It gives you an added clout."

Feinstein sips her coffee and stretches her legs out toward the lake, which looks sensational because the sun is shining on it and also because the parks department, with the encouragement of the mayor, is conducting a major cleanup of the parks. She pointed this out as she jogged toward the lake, trotting slowly and with great purpose down the narrow paved paths, her body tipped forward, her head bent in concentration. Nobody recognized her. Nobody recognized her at first at the Stow Lake coffee shop, either. When she jogs with her vigorously social friend Charlotte Maillard, people sometimes pass them and say, "Hi, Charlotte."

It is not that nobody knows the Feinstein face--hardly that. Feinstein was in San Francisco public office before most of today's junior high school students were born. The problem here is image, which for years has enveloped Dianne Feinstein as surely as a phalanx of fat bodyguards. If she jogged in a well-cut suit, with low dark pumps and a bow draped over a high-necked blouse, then people would know. They would nod as they passed. 'Morning, Mayor Feinstein.

" 'Goody Two-Shoes,' " Feinstein says, sounding more wry than irritated. " 'White Gloves.' " What else? " 'Pacific Heights Matron.' " She winces. "I've never even felt a matron. To me a matron is a big, buxom woman--" Feinstein laughs out loud, her hands carving an enormous bosom in the air--"locking the door. Has keys dangling from her belt."

But of course that is not a Pacific Heights Matron at all, not in San Francisco, where neighborhood clings to you from childhood and blurs, or so the city likes to suppose, your lifelong vision of the world. There were some boys, a while back, who came from outside Pacific Heights to go to school inside Pacific Heights, and when they were on their own they would talk, their voices dark with snubbed contempt, about "94115 girls." So inviolable are the myths of territory here that you can repeat this label to almost any native San Franciscan and he will understand instantly what it means: within the ZIP code boundaries of 94115, inside those splendid brick and marble constructs with dizzying views of the bay, lived imaginary Grace Kelly sorts of girls, all expensive and patrician and aloof beyond reach.

Dianne Feinstein was first elected to the Board of Supervisors in 1969, and it was thereafter her privilege and misfortune to emerge in San Francisco politics as the quintessential 94115 girl, the Pacific Heights supervisor, the reserved, even-voiced woman whose precisely cut suits and wind-resistant hair style seemed to satisfy every local conceit about wealthy women who live astride that one coveted hill. She kept her private life private and worked like a fiend and never talked publicly about how she felt, and to this day, all over San Francisco, the men and women of the less moneyed neighborhoods will say that Pacific Heights is how Feinstein thinks about things, that Pacific Heights made Dianne Feinstein what she is.

But it didn't.

She is tired of hearing about how easy she's always had it.

Here is part of what made Dianne Feinstein what she is:

Her father was a surgeon, Jewish, gentle, protective. Her mother was a great beauty, Russian Orthodox Christian, an immigrant, and vicious.

"Capable of great hostility--I mean, terrible things," Feinstein said about her mother in a local magazine interview last month. It was an extraordinary interview, in which Feinstein talked publicly, for the first time, about growing up with a mother who had apparently suffered brain deterioration in childhood as a result of encephalitis. The disease had left her mother irrational, Feinstein said in the interview, the kind of woman who would pull her daughters' hair and paddle them and throw Dianne, the oldest of the three, out of the house now and then.

"A role falls to you, whether you like it or not, and that's really your beginning with leadership," Feinstein says. "So you learn--like if you've got to cook the breakfast, and you've got to see that your sisters eat it, and you've got to see that the house is clean, and you've got to do some of the shopping--you're going to have to learn, in a very elemental way, certain ways of getting all this done. Plus, if you're smart, getting them to help you do it."

There is a drive, in differently besieged families, that sometimes consumes the oldest children of alcoholics, Feinstein is told: a passion for self-control, a push for achievement, a need to hide the secret and at the same time work furiously, and usually in vain, to fix it. The problem in her family was not alcoholism, but--

"Oh, yes, it was," Feinstein says. There is no audible emotion in her voice, and her gaze is close and direct. "If you're not dealing with a whole brain, a lot of things happen. Alcohol was one of them. I approached it in two ways. I'd empty bottles and fill them half with water--there's a fine balance you can get to before it's known. And continual attempts at suicide was another problem, and having to handle that. I mean, that's the tough part. The inordinate fear when you're a youngster and you see your mother, um, you know, being walked--my father of course was a doctor--filled with pills, and forced vomiting--all the stuff that goes with it.

"And when you're young you're shipped off to friends while it goes on, and then at one point you're a part of it," Feinstein says. "It's a very frightening thing." Mayor With a Memory

On Monday morning at 9, three rows of folding chairs face the massive wooden desk that, as Feinstein likes to observe, once belonged to Franklin Delano Roosevelt. The desk is hers now, and she steps from behind it to greet the department heads rapidly filing in over the huge oriental rug. There are women in the room, but only a few; it is a weightily male congregation Feinstein faces, and she moves easily among them, patting shoulders and leaning in for quick low consultations, her jacket and bow a startling blue amidst the press of dark business suits.

"If you don't mind, we'll begin," Feinstein says, and from there it is brisk and direct, the regular Monday reports. The mayor quizzes, nods, makes notes on small cards filed inside the kind of large leather volume that airline magazines advertise to "organize the busy executive." Feinstein has the good politician's eye contact, the intensity of gaze that seems to drive all others from the room, and she shifts it effortlessly now from one face to the next, requesting this, acknowledging that. A church needs city help finding a new site so a park can be built on the old one; Feinstein wants the new site appraised sooner to get this moving. "Do you think you could have something next week, something specific?" she asks the administrator. She asks about the projected color of a building due for rehabilitation--terra cotta with blue trim, is that right? She asks the police chief whether he might station his litter control unit at the downtown cable car turntable to keep the street corner cleaner. She asks the Housing Authority director for a due date, in writing, on a new housing project--"one that you are prepared to stake your career by," Feinstein says.

She smiles as she says this, but the humor lays a very thin veil over the Feinstein managerial style: a fervor for order, a direct involvement in the specifics of city government, an attention to detail that has been the pride and despair of her staff. These are people who know how she operates, who watched her come into office and personally review 200 routine job vacancy forms a week, who know she will press them and keep after them and want things in writing, rewritten again and again if necessary--that the mayor will battle bullheaded against what she calls the scourge of "Aha, she'll forget."

So there is laughter in the room, but it is not the kind of laugh George Moscone would have gotten. George, as he is invariably referred to by people fondly invoking his memory, was such a different sort of mayor, a man whose administrative abilities were greatly exceeded by his charm. Minorities liked him; women liked him; people who had traditionally been kept from power liked him.

The day he was shot to death, in a corner of his own office, it was Dianne Feinstein who found the body.

Her voice, in the terrible memory of that week 5 1/2 years ago, seems entwined somehow with the sound of Joan Baez, who kept appearing in public places and singing "Amazing Grace" and whatever music of solace might pull people along from one day to the next. The whole city was benumbed already by the Jonestown deaths, and when just over a week after Jonestown the news came crashing down from City Hall--that the conservative former supervisor Dan White, ostensibly in a rage over Moscone's failure to give him back the supervisor's seat White had quit, climbed through a basement City Hall window and shot to death both Moscone and gay supervisor Harvey Milk--then, for the people who kept gathering in quiet, desperate memorial services, the thing was nearly too much to bear.

And Feinstein kept saying it would end, that it would be over, that the city would go on. The Board of Supervisors made her mayor, and from the moment she had to walk into the City Hall corridor to tell the small assemblage of reporters that Moscone and Milk were dead--the tapes, played again and again in the aftermath, recorded some reporter's full scream--Feinstein did what she had to do with such grace that even her most ardent political enemies soften still when they remember it. "It is my duty to make this announcement . . ." She stood up straight and sounded steady and yet compassion somehow resonated from her every time she spoke, or moved among a grieving crowd. "As we reconstructed the city after the physical damage done by the earthquake and fire, so too can we rebuild from the spiritual damage . . ."

She had run for mayor twice in her career, and been beaten both times so badly that the severity of the trouncings astonished her. She had convinced herself that she was unelectable, that it was time for her to leave city politics. And now, in a city cracked by death, with a massive antidiscrimination suit facing the police department and a $130 million budget deficit brought on largely by the tax-cutting initiative Proposition 13, Dianne Feinstein was the mayor of San Francisco. The Devastating Win

She still winces when she hears the complaints about her. "It's damn tough being mayor," Feinstein says with sudden vehemence, jogging slowly from Stow Lake into the park. And earlier, in her office, the decorum more securely in place: "The mayor, in my opinion, that's going to please everybody, and is going to be extraordinarily popular, is going to end up being a very bad mayor. It just isn't possible . . . At the recall, when I was out at the supermarkets, I ran into three people who had the guts to come up to me and say, 'You know, I gotta tell you, I signed the recall petition.' "

The mayor squints sideways, as though peering at these hostile constituents. "And I said, 'Do me a favor--just tell me why,' " she says. "One said, 'Because I can't find a parking space anywhere in this town, and I'm angry.' The second was, 'I waited and waited 45 minutes for the city bus, and it didn't come.' And the third, um--it had something to do with the sidewalk . . . You're the person who gets blamed for everything."

What the people who get mad at Dianne Feinstein say is this: That San Francisco is being consumed by enormous financial district high-rises that look hulking and bring in unmanageable numbers of commuters. That the city is approaching Manhattan in its wildly expensive housing, with one-bedroom apartments renting for $800. That Feinstein is too chummy with the developers and financiers whose names are generally invoked whenever people cry that the city is economically shoving out its own middle class.

They say that Feinstein, the elected head in this city of such celebrated liberalism, is a disappointingly staid politican. She supports rent control, but refuses tenants' pleas to extend controls to vacated housing (makes builders reject new construction, she says). She lobbied for the Equal Rights Amendment, but refused to have staff members on city business join a boycott of states that had not ratified the ERA (counterproductive, she says). She has not actively opposed organized labor in this strong labor town, but has made no effort to embrace it; she is accused of appointing too few minorities to commissions; she had close ties with gays for years and then enraged them by vetoing legislation that would have allowed residents to obtain some spousal rights for their lovers by officially declaring them "domestic partners."

Hardly anybody argues about the list of Feinstein accomplishments--the crime rate is down, the city government purrs along under closely monitored new management techniques, the local unemployment rate dropped to 7.2 percent this year, the downtown shopping area thrives, the city just weighed in with a $130 million budget surplus, and last Thursday, in a massive urban celebration culminating two years of costly underground repairs, the cable cars came back. So when Feinstein learned in early 1983 that some Haight-Ashbury neighborhood activists had begun circulating petitions for her recall, her staff initially dismissed them as crackpots; their chief complaint was the mayor's support of a law, since overturned by the courts, that prohibited pistol ownership inside city limits.

"And then--fwooom," Feinstein says, laughing, her arm arcing down in a great imaginary cascade of paper. "In came the petitions!" City law required just over 19,000 signatures to order a recall election; the petitions came in with nearly twice that many. In April 1983, seven months before the regularly scheduled mayoral race, Dianne Feinstein faced a special election to see whether she could keep her job.

Her friends uniformly say the recall nearly took her apart, and she agrees. "It was like a knife was going in," Feinstein says. "I felt a deep sense of personal humiliation that after, what was it, 14 years in public life, finally getting to be mayor in my own right--I don't think anybody has ever said I don't work hard; I work very hard--people wanted to throw me out of office. That's what it meant . . . They were saying, like in a ballpark, 'You're a bum, and we want you out.' And it hurt."

She won the recall. She won a devastating victory, commanding 81.2 percent of the vote, and at a press conference the next day Feinstein declared in an intemperate moment that the November election was now entirely hers. "Anybody thinking of running is going to be creamed," she said.

She was right. The mayor's opponents that November, all of them dispensed with in another Feinstein landslide, included two women socialists, a nightclub owner and a self-described "economic theoretician" who said he believed homosexuality could be cured. And something had happened to Feinstein in the course of the recall: forced to campaign in the streets with a vigor she had never summoned before, and greeted by thousands of people who thought the recall was a terrible idea, Feinstein found her legendary public reserve begin to give way to a genuine pleasure at political crowds.

"I used to be very apprehensive," Feinstein says, gazing out the limousine window now as the black Lincoln carries her home from the park. "I suppose the newness of the job, the apprehension, the kind of anxiety of--well, you know, maybe people don't respond to me, or like me." Now there is no more apprehension, she says. "I don't know a second-term mayor who has that 81 percent vote." Toughing It Out

It was her Uncle Morris, she likes to say, who showed her politics. She was a San Francisco teen-ager (who never even moved to Pacific Heights until the seventh grade); he was a women's clothing manufacturer who ran a very early television talk show, took Dianne to political meetings and walked the streets like a seasoned ward heeler. "I loved my uncle very, very much," Feinstein says. "He taught me everything from don't-lick-your-knife-at-the-table to being able to discuss--rip-roaring discussions at the table."

The young Dianne went to Sunday temple school, and when the two secular Pacific Heights private schools denied her admission--because she was Jewish, she believes--Dianne Goldman became the only Jewish student at the Convent of the Sacred Heart high school. She went to Stanford University, abandoned the idea of becoming a doctor like her father, ran successfully for student body vice president and graduated to work in criminal justice. Then-governor Edmund G. Brown, on the basis of a Coro Foundation report she wrote about the criminal justice system, appointed her to the California Women's Board of Terms and Parole. Feinstein married, had a daughter, was divorced from her husband; in 1962 she married Bertram Feinstein, a brain surgeon who was 19 years older than she was.Bertram Feinstein died of cancer six years ago, while Dianne Feinstein was in her third term as a San Francisco supervisor. She had made a spectacular entrance into city politics, defying conventional men-only wisdom and attracting more votes than any other 1969 candidate, but by the time her husband died she was exhausted and worn raw by his cancer. Did she think about counseling? "I am of the school that toughs it out," Feinstein says. "I didn't really feel like I needed therapy. I mean, this was, um, one of those--"

Her voice breaks. "I still can't talk about it," she says.

There is a bad moment, and Feinstein looks away, her face working. "Oooh," she says finally, sounding startled. "Let me kind of get back together." She lifts her sunglasses and wipes a hand across her cheekbone. "I guess part of what you do is you mask grief, and you can do it. But it comes out at--oooh. I'm sorry."

Feinstein takes a deep breath. "You know, what I really learned--I think the human things that people go through are by far the most important things in life. And they're the most important things to respond to."

That is what people say about the private Dianne Feinstein, even if they complain about her policies in the next breath: She is passionately loyal, capable of sudden wicked humor and possessed of a fierce, practical generosity in times of personal crisis. A city sheriff's office employe still remembers from many years ago the way Feinstein, who had intercepted the call bringing terrible news, took her out for a drive to tell her with infinite gentleness that the woman's husband had died of a heart attack. When the daughter of one of her deputy mayors was killed recently in a car crash, Feinstein was nearly the first person on the telephone to him and arrived at his home laden with coffee cake to serve the visitors she knew would begin to arrive. And when one of her fire chiefs was hospitalized after a bad on-the-job accident, Feinstein went straight to the hospital and then headed out to find an all-night drugstore so she could bring a comb and toothbrush to the chief's distraught wife.

"I know when I was sick a couple of times, you know, the fact that somebody would call means a lot," Feinstein says. "It keeps the human perspective that I really need to keep in this job, because the office part of it drives you to distraction. You know, you feel yourself very tough, and hard--"

Feinstein punches the air, her voice going mean. "And I hate it," she says. "Just to--just to move--the most intransigent thing is a bureaucracy. People figure they're going to outlast you, they're going to be there longer, they're not going to pay much attention. And you've got to break through that . . . part of the exasperation is you can talk about something for a year, you know."

Mocking now the polite tones of a reasonable administrator: " 'This is what I advise, this is what I want.' And then it comes back--" she waves her arm extravagantly and bursts into laughter--"totally the wrong way!"

And if you are a woman, you must, of course, yell twice as hard and look twice as mean, until after a while they get used to you, Feinstein says. If you have been mayor for 5 1/2 years by then and if your name is mentioned in the same sentence as "vice president" by people not simply out to flatter, then the pressure to be tough starts at last to ease off. "I'm finding it less and less now," she says. "I think I've made the point."