At 68, he is the oldest governor in the country. It is 28 years since he first ran for the office, and he has held it for 12 of the last 16 years. The state's biggest city is teetering on the edge of bankruptcy. School districts across the state have been closed by fund shortages and teacher walkouts.

He has been feuding with the state legislature, whose House minority leader is challengng him in the June 6 primary on the grounds that "his lack of leadership has made Ohio the crisis capital of the nation."

His press relations are equally stormy. Recently, statehouse correspondents formally petitioned him for a press conference, after a two-month lapse.

He has refused repeated demands that he make a full disclosure of his income and assets. Yet it is known that he converted at least $54,000 of campaign funds to his personal use. In the last 40 years - all but four of which he was on the public payroll - he has become a millionaire from real estate and franchising interests.

On occasions when reporters have not been able to find him on his office, they have discovered him at sales meetings of one of his business interests or at the Florida consominium where his wife spends most of her time.

Surprisingly, in this post-Watergate era of American politics, Gov. James A. Rhodes is considered a strong favorite for reelection.

The reason, says State Rep. Charles F. Kurfess, his challenger in Tuesday's Republican primary, is that Jim Rhodes is a unique kind of guy. He has a innate political sense of the bottom line."

He's a genius at seizing a crisis and either capitalizing on it or putting the blame elsewhere," echoes Ted Celeste, brother and campaign manager of the Democratic challenger, Lt. Gov. Richard F. Celeste. Celeste has only token opposition Tuesday from Dale Reusch, a wizard in the Ku Klux Klan.

The frustration both Kurfess and Celeste express in coming to grips with the Rhodes phenomenon is deepened by the elusiveness of their prey. He is everywhere, in terms of name recognition, and yet he is nowhere, so far as his actual participation in the campaign is concerned.

Having been on the ballot in almost every possible election since 1952, when he was first elected state auditor, Rhodes is literally a household name in Ohio. But his only campaigning so far this year is at fund-raising breakfasts.

He has printed no literature nor made any television commercials against Kurfess, who is dropping $120,000, mostly family money, into a television drive for an upset.

Rather, Rhodes is sitting on a $731,000 kitty for the fall campaign, which will undoubtedly be used for a television biltz similar to that which carried him to a comeback victory over Democratic incumbent John J. Gilligan four years ago.

Rhodes had enjoyed two succesful terms as governor from 1962 to 1970, emphasizing jobs and economic development with the slogan. "Profit is not a dirty word in Ohio." Teams of businessmen, called "Rhodes' Raiders," lured new industries by bragging about the state's low tax burden.

But by the end of that second term, some of Rhodes' magic had worn off. Life magazine had published charges about his private finances and a grand jury had investigated his role in the Kent State tragedy. Barred by law-from seeking a third consecutive term, he challenged Robert Taft Jr. in the Republican senatorial primary and lost.

For the next four years, Rhodes devoted full time to his business interests and his love of golf. But in 1974, he staged a comeback against Gov. Gilligan, a Democrat who had pushed through the state's first income tax law.

In an effort to get off the defensive on taxes, Gilligan challenged Rhodes to debate and to disclose his personal finances. Rhodes - who had done both when he lost to Taft - declined, and instead blitzed Gilligan with a television ad campaign which, ironically, featured the threat of padlocked schools. He won by 11,000 votes, in the face of a statewide Democratic land slide, leading Gilligan to comment, In the year of Watergate, I lost to a moral paraplegic."

The last four years have been rough for Ohio. School closings have spread, with no less than 116 districts facing, shutdowns in Septmeber unless levy increases are approved Tuesday - a doubtful prospect in many cases. Cleveland has a financial crisis that rivals New York City's. Despite Rhodes' success in luring two new auto plants into the state, manufacturing jobs have declined sharply. Natural gas and coal shortages hit Ohio harder than almost any other state in the last two severe winters.

But, as Gilligan comments, "things that would seem to cripple the ordinary politician just bounce off him (Rhodes). He is a master at giving the general impression of being in charge of everything - and responsible for nothing."

Thus, during the blizzards of 1977 and 1978, Rhodes' reassuring voice was heard constantly on the air waves.

This year, no one has been move vocal than Rhodes in blaming the Democratic legislature for the school-finance crisis or the Environmental Protection Agency for the shutdowns of steel plants in Youngstown.

The governor declined The Washington Post's request for an interview as he does most newspapers. But Robert Teeter, the Detroit pollster who handles Rhodes' campaigns, probably gives a clue to his survival skills when he says. "He has as much sensitivity and perceptiveness about the average voter as anyone I've ever seen. He can sit in his office and know what that auto worker in Elyria is thinking."

Kent McGough, Rhodes' campaign manager, describes the governor as 'very pragmatic. He brings it down to something people can understand. He says, for instance, that he stands between the taxpayers and the taxspenders, and they understand that. The bottom line is that he comes through as a guy who's interested in people and their problems."

It is that special Rhodes aura of blunt-spoken common sense - expressed forcefully if not always gramatically - that Kurfess is struggling to shatter in the primary and that Celeste will probably be up against in the fall.

Kurfess, who is 20 years Rhodes' junior, tells Republicans that he supported the governor in his earlier terms, but "the last four years he's just given no leadership. This state can't afford to drift any longer."

Kurfess has issued detailed policy papers on all the issues he accuses Rhodes of ducking, and - like others before him - has futilely challenged the governor to disclose his personal finances.

But his effort is underfinanced and, with the Republican organization clearly in the governor's corner. Kurfess' personality has failed to ignite a strong insurgent movement within the GOP. Most observers guess he will do well to get between 30 and 40 percent of the primary vote.

Celeste is - at least on paper - a much more formidable challenger to Rhodes in the fall. A 40-year-old grandson of Italian immigrants to the Cleveland suburb of Lakewood, a Yale honor graduate. Rhodes scholar and a Peace Corps veteran. Celeste cultivates a Kennedyesque speaking style - and, some say, the ambition to go with it. He won by 217,000 votes for lieuenant governor at the same time Rhodes was beating Gilligan.

He avoided serious challenge in the primary and has a relatively strong and united Democratic Party behind him, despite the bruised feelings left behind in his rapid rise to the top. His young and dedicated personal organization has been pointing for this gubernatorial campaign for four years and President Carter, Vice President Mondale and Sen. Edward M. Kennedy (D-Mass.) are all lined up for fall fund-raisers that will benefit Celeste's drive.

Celeste says that if Democratic unity holds, he can make Rhodes the issue in the fall - and beat him on his record. But he does not underestimate the problems involved in what his campaign manager and brother, Ted, calls "running against an institution."

Despite his busy speaking schedule as lieutenant governor. Celeste is still an unknown to most Ohio voters. He concedes he is still grasping for a way to "crystallize the feeling that Ohio is slipping" into an effective argument against reelecting Rhodes.

"I know what he'll do," Celeste says. "He'll hoard his money and then dump it into television in the last four weeks. He'll ignore me and what I'm saying about his record. He'll run against Jimmy Carter and the federal bureaucracy and Jack Gilligan and the liberal spenders - and suggest that I'm somehow linked to all of the above.

"It's always an uphill battle against a guy like Rhodes."