MOSCOW, FEB. 20 -- Elena sleeps in the train station. She is 37, but the life she leads has etched another decade into her face. "This life without a home, it's impossible," she says. Sometimes, when the police chase her out of the station, she finds her way to an airport and spends the night there. In the summer, she says, the stations are too crowded and the city parks are the place to go.

"I'm homeless for eight years," Elena says, sipping a glass of tea in a cold, grubby cafe. "Sometimes I get 5 rubles a day scrubbing the floors on the trains after they pull into Moscow. Right now I'm broke and everything I have is what you see -- the coat and the clothes I'm wearing."

Across the table sits Leonid, a huge, handsome man in his mid-fifties. When he lost his job five years ago as an economist in Leningrad, his wife left him.

Now Leonid wanders, picking grapes in the summer to make some money, sleeping in bus stations, breaking into empty summer cottages when no one is around. "I live on 50 kopecks a day," he says. "I get by on tea, some porridge and milk, that's all."

Elena and Leonid -- both of whom preferred that their last names not appear in this article -- have sent countless letters to the Soviet government, and they say that many of their homeless friends have written, too.

"I've written Mikhail Gorbachev, Andrei Gromyko, everyone," Leonid says. "I want my right to work and live guaranteed by the constitution of the Soviet Union." Elena nods and says, "You know, there are thousands like us in this country. Thousands."

For years, the Soviet press has printed stark pictures of the American homeless, desperate souls stretched out freezing or dead on street corners and subway grates. A couple of years ago the Soviets tried to score propaganda points by filming a television documentary about the life of a homeless man who slept for years on the sidewalks of New York and then enjoyed a first-class trip to Moscow, courtesy of the Soviet Union.

Now the official press here is beginning an inward search, looking at their own homeless, at street people known by the acronyms bozis and bombzhis -- people like Elena and Leonid. Bozis, or people "without definite direction or work," and bombzhis, or "people without a definite place of residence," are a growing problem in the Soviet Union, especially in southern cities, where the homeless try to escape the long, northern winters.

Two weeks ago, Leonid Kirienko, a self-described tramp from the Ural city of Sverdlovsk, wrote a desperate letter about his plight to the editors of Izvestia. "Judging from the articles in our newspapers," he wrote, "homelessness only exists abroad. But in our country there is enough of it."

"That letter was a real first for us," said Izvestia's deputy editor-in-chief, Nikolai Efimov, in an interview.

Kirienko described how he was out of work, how for years he has lived the life of a tramp. "I do not have a return address," he wrote.

Under former Soviet leader Leonid Brezhnev, such a daring letter on a social problem would have been a sensational lapse into honesty. But in the era of glasnost, such signed complaints, no matter how brief, tend to signal that a discussion of a particular problem in Soviet society is about to become public.

Indeed, a fuller treatment of the homeless problem was not long in coming.

"We must speak as loudly and openly about vagrants as we have about drunkards and dope addicts," wrote Izvestia correspondent Albert Plutnik in a long article this week. "It too is a social evil, our new, unexpected social problem. It should not have happened, but it has. We must publicize it and open our eyes to it."

So far neither the Soviet press nor the government has published statistics on the problem; therefore there is no way to estimate the scale of homelessness in the Soviet Union. But if recent discussions of other social problems, such as prostitution and drugs, are any indication, it is likely that the press will soon paint a clearer picture of the life of bozis and bombzhis.

In Moscow, homelessness is nearly invisible. Moscow is the showplace of the Soviet Union, the home of the nomenklatura, or Communist Party elite, the intelligentsia, the relatively wealthy. Like nearly all urban Soviets, Muscovites suffer from a lack of living space, often sharing tiny apartments and communal bathrooms and kitchens with their extended families and neighbors.

And yet no one sleeps or hustles change on Kalinin Prospect the way people do on Connecticut Avenue in Washington. The Moscow police are too numerous, too vigilant for that. According to Muscovites, when destitute people show up at city train stations, they are sent away.

Elena shows her passport and says, "You can see I don't have a real propiska, no permission to live in Moscow, and you need that."

In the Izvestia article, Plutnik described the problem as it exists far outside the capital and displayed a mix of sympathy for the homeless and a firm conviction that most of them have wound up in such dire condition through no fault of the state.

"As winter approaches, like migrating birds they are drawn to the south," the article said. "There it is warm and easier to find something to eat.

"Some of the migrants head straight for the Tashkent militia station, flowing from all corners of the country. They give themselves a month off. That's how long the militia will keep them to find out their identity. During this period they are guaranteed a roof over their heads and free meals. Afterwards they will be chased off anyway. Parasites can't be maintained on state food."

Plutnik described the refugees at the militia station in Tashkent, the capital of the Soviet Central Asian republic of Uzbekistan: a despairing man who left home after discovering his wife had been unfaithful, an old, senile woman who one day walked out her door and never found the road home -- and an "abnormal" young woman who thought she was the deposed queen of Iran.

The article told the story of Pyotor Maslov, a wanderer in Uzbekistan who was beaten one night by teen-age thugs who wanted to steal his wine. A man found Maslov unconscious in the street and brought him home, to discover that Maslov had no job, no pension, no wife and no idea of his son's whereabouts.

"Who is he, this Maslov?" Plutnik asked. That plaintive question seems to be a starting point for a Soviet discussion of a troubling phenomenon, an expression of wonder about how Soviet citizens could possibly slip through their vaunted safety net of cheap housing and guaranteed work.

"To what social group or stratum does {Maslov} belong?" Plutnick wrote. "He's a vagabond who lives here and there . . . a bomzh, a boz."

The article concluded sentimentally, with the author tracking down Maslov's 32-year-old son and the son vowing to bring his father home and care for him.

Recent debates and articles over prostitution, drug addiction and alcoholism suggest that future discussions of the problem could be far more penetrating.

The Armenian Communist Party journal Kommunist last week published an article claiming there were 362 vagrants in that Soviet republic, on the Soviet Union's southern border. The article described them as "disheveled, unshaven, wearing tattered clothes, sunglasses and caps with ear flaps." The paper said, "They can be seen in underpasses, public transportation and at cemetery gates."

Unlike the relatively sympathetic article in Izvestia, the Armenian paper warned against criminality, prostitution and venereal disease among the "vagrants."

"Large numbers of these degenerate people are from bad families, runaways from orphanages, young people who could no longer stand their parents' beatings and drinking bouts and fell under someone's bad influence," the paper said.

The problem of poverty in general may not be immediately evident in the wealthier, tourist cities of Moscow, Leningrad or Kiev, but it is present. According to a recent study, "Poverty in the Soviet Union" by British scholar Mervyn Matthews, two-fifths of the non-peasant labor force earns less than the amount of money that Soviet scholars say is the subsistence level for a small urban family. Poverty among retired people, peasants and larger families, Matthews wrote, is "undoubtedly higher."

Many here believe that Gorbachev's economic reform plans, which emphasize efficiency, are bound to lead to the "capitalist" phenomenon of unemployment, or "reductions" as the official announcements and speeches politely call it.

In a letters column printed in Izvestia the same day as the story about the homeless, A. Salhodyayev wrote, "Is it really our understanding that unemployment will soon become part of our daily lives?" That is the fear among many workers, and if unemployment increases sharply, analysts say, the number of homeless could rise as well.

One western diplomat here said his embassy receives "hundreds" of complaints from homeless people who see western embassies as a kind of last resort.

"Most of these people come from outside Moscow and they try to present their case at the reception office of the Presidium of the Supreme Soviet," the diplomat said. "So what happens? They spend five minutes talking to some bureaucrat through a little window and then the bureaucrat says he can't do anything to help."

The diplomat said he has seen more homeless in the last eight months. "I'm not sure if it's a matter of glasnost and people being more willing to speak their mind or if it's a rise in unemployment," he said.

Elena, who said this morning that she will spend tonight sleeping on a bench in a Moscow train station, said she really only has one question for her government: "Where am I supposed to go?"