Doris Rizkallah's coworkers in the mailroom of a Gaithersburg company have no idea that she is homeless.

"It's not the kind of thing you tell people," said Rizkallah, 28, a baby-faced woman dressed neatly in a gray plaid skirt and black pumps. "I listen to other people talk about their new homes, and they just bought a car, and you see them wear different clothes every day. Sometimes it gets pretty hard."

Since early January, "home" for Rizkallah and her three daughters has been the Stepping Stones shelter in Rockville, a sprawling, renovated farmhouse tucked in a development of $95,000 brick town houses.

Rizkallah is one of the hidden homeless, the growing number of individuals and -- increasingly -- families who find themselves without a place to live in the Washington suburbs.

Some of those who are homeless in the suburbs fit the stereotype of the more visible, and more numerous, homeless downtown -- the deinstitutionalized, released from mental insititutions with nowhere to go, and the chronically homeless, often suffering from alcoholism or drug addiction.

But a greater number are what Jill Lienhardt of Bethesda's Greentree Shelter calls "the new homeless," who never expected to find themselves in such desperate straits.

Often working at service jobs for the minimum wage of $3.35 an hour, or living on public assistance checks, they may be hit by a run of bad luck, or their housing situation may simply collapse under the strain of skyrocketing suburban rents.

With waiting lists stretching to as long as seven years, low-income housing offers little prospect of relief. As a result, the homeless often resort to lies, inflating their income to prospective landlords in order to rent an apartment or hiding part of it in order to keep their welfare checks from being reduced.

The Rizkallahs' problems began when Joe Rizkallah lost his job working with his brother as a painter, and he was unable to get unemployment benefits on the grounds that he was self-employed. Doris Rizkallah's job at the time, earning minimum wages by typing and filing for a local engineering firm, did not bring in enough to pay the $545 monthly rent. Evicted From Town House

She came home from work one October afternoon to find an eviction notice tacked to the front door of their three-bedroom town house in Gaithersburg.

"I heard about people sleeping in the streets downtown," said Rizkallah, who has since separated from her husband. "It wasn't a reality until it happened to me."

"I have to keep reminding people that there are homeless people in the suburbs," said Robert Crittenden of Catholic Charities, which operates five suburban shelters in Maryland and Virginia. "They say, 'Where are the bag ladies, where are the people we see sleeping on the grates downtown? No grates, no people.' But they are there."

Fairfax County officials, for example, estimate that there are 390 people homeless there on any given night, and that is a conservative figure, according to Donna M. Foster, assistant director for adult and family services. In Montgomery County, between 800 and 1,000 people are homeless, according to social service officials, and in Prince George's County, between 1,000 and 1,200.

"Many of the people that end up in the shelters have jobs and are different from the people who sleep on grates downtown," said Lynda Eubank, assistant division chief for the Arlington County Department of Social Services. "There are more and more people who would have been able to make it and are now bumping into the edge of survival. They were barely making ends meet before and now they find they can't do it." Problem Is Nationwide

The story is the same in suburbs across the country that, once thought to be affluent enclaves immune from such urban problems, report increasing numbers of people with no place to live, despite the economic recovery.

"It's less visible . . . . It's not going to hit you in the face the way it would walking the streets of New York City," said Dan Salerno of the National Coalition for the Homeless, a New York-based organization. But, he said, "The suburbs are noticing it, and to me that's an indication that it's getting worse."

A growing number of the suburban homeless are families, both intact and single-parent. "We are seeing a new need -- families, often single moms," said Gary Cyphers, Alexandria's assistant city manager for human services.

In Prince George's County, for example, a survey by shelter providers found that 77 percent of their requests for housing aid came from families with children, with an average of two children per household. Nearly two-thirds of the cases involved single parents; in half, the primary source of income was public assistance or unemployment benefits.

Added Jana Graves, director of the Mondloch House shelter in Fairfax County, "They'll be at the shelter, and he'll be at a night job and she'll be at a day job and they'll have two kids and they just won't be able to make ends meet . . . . People who used to be able to live in this area on public assistance can no longer afford to. They're going backwards like 100 miles an hour."

County officials and shelter providers throughout the Washington area warn that the problem is getting worse:

* Crowding forced the Arlington Community Temporary Shelter to reject 298 people in the last half of 1984 -- five times as many turned away in the previous six months, and seven times as many as in the same period of 1983.

In an effort to serve more people, the shelter, starting Jan 1, also cut the maximum stay from three weeks to two. "How could you and I find a job, get a place to live, deal with everything in two weeks?" asked shelter director Sheila J. Wolfe. "It's not a good solution, but I can't think of any other."

* At Shelter House in Falls Church, the average length of stay has climbed steadily, from nine days in 1982, to 16 days in 1983, to 23 in 1984. "It will probably be close to 30 this year," said shelter director Jane Deppler. The shelter turned away 68 people last September, she said, three times the number in September 1983.

* Prince George's County sheltered 450 families and individuals in local motels and churches in the fiscal year ending in June 1984, compared to 320 in 1983 and 301 in 1982. County officials expect to shelter more than 480 individuals and families this fiscal year.

* Warning that costs for their homeless program could be double initial projections of $20,000, Alexandria officials were forced last December to seek additional funding for the city's emergency shelter program.

"We had one family going out today and we had two waiting to go in," said Gus C.E. Hall, director of Alexandria's Christ House shelter. "Then we had a vacancy for one man and we had several apply."

* Montgomery County paid for housing in local shelters and motels for 167 families and individuals in 1982, 223 in 1983 and 270 last year. Those figures do not include the growing numbers who received free shelter from church groups and other organizations in the county.

* In Fairfax County, costs for sheltering the homeless have risen from $168,000 for 801 people in the fiscal year ending in June 1983 to $301,000 for 1,133 people last fiscal year. Costs this year are expected to exceed $500,000, said Foster.

"We've been running at 100 percent occupancy for six months," said Graves of the county's Mondloch House shelter. "We have 32 beds, but I often go over that. It's hard to turn people away."

Typically, homelessness is not a sudden occurrence but a gradual descent. Evicted, or on the verge of being evicted, an individual or family may move in with friends or relatives. But often, either landlords discover the doubling-up and order the family to leave, or crowded conditions breed tension and arguments, and the family finds itself without a place to stay.

That leads them to shelters or to cheap motels that counties use as an alternative to shelters, either because they find motels more "dignified," as Prince George's County, or because existing shelters are filled to capacity. Shelter Stays Are Limited

Since many shelters limit stays to a fixed period, the homeless often find themselves on a round of "musical shelters," simply moving to a different shelter without being closer to resolving their housing problem.

And once they have patched together a solution -- finding another family to share quarters with, stretching a tight budget for an apartment above their means -- they may find themselves homeless once again.

"They get it together and manage for a few months and then get behind again," said Eubank of Arlington. "It used to be in cycles of a year or two years," she said. "Now it's a matter of months."

The essential problem is that the rise in housing costs -- fueled by a shrinking market of available apartments and heavy demand as the metropolitan area population grows -- has far outstripped the increase in income of many suburban residents.

Among the Washington suburbs, only in Prince George's County is the average rent for a one-bedroom apartment below $400 monthly; a mother with two children on public assistance, in contrast, receives a maximum payment of $363 from the Aid to Families with Dependent Children program.

"No matter what kind of documentation you use, you keep coming up with the same conclusion: that, especially in the suburbs, there is a terrible shortage of affordable housing," said Barry Zigas of the National Low-Income Housing Coalition. High Rents Cited

"Just to move into . . . a decent, clean apartment in the Rte. 1 area will take $800 to $1,000 in up-front money to pay the security deposit, pay the first month's rent, get the utilities turned on," said the Rev. Vin Harwell of Mount Vernon Presbyterian Church, one of nine Fairfax churches in the Rte. 1 area that take turns sheltering homeless men and women. "Where are they going to come up with that kind of money -- and if they do, what are they going to live on to put food on the table? When you're washing dishes part time at Denny's you can't afford $350 or $400 a month."

Moreover, most apartment complexes require that potential tenants earn at least 75 percent of the monthly rent in a week -- a standard that few of those who find themselves homeless can meet.

While some of the suburban homeless move to the District, which has a more plentiful supply of cheap housing, many are fearful of leaving the suburbs, which they consider safer and as having better schools. Others want to live in suburbia because that is where their jobs are.

Consequently, waiting lists for low-income housing in the suburbs stretch into the thousands, except in Alexandria, which stopped taking applications three years ago and still has 600 people on its waiting list.

The scarcity of subsidized housing is expected to worsen. During its first term, the Reagan administration virtually halted new construction of subsidized housing -- the full impact of which has not yet been felt, according to Zigas. Now, it is proposing a two-year freeze in the number of families receiving housing assistance.

"You can have all the temporary resources in the world," said Helayne Baker of Stepping Stones, which gets its funding from local churches, synagogues and individuals. "If you don't have those permanent resources -- like affordable housing -- it doesn't make a blessed bit of difference."

"We're in for a siege of homelessness," warned Cyphers, Alexandria's assistant city manager.

After the eviction and the Rizkallahs' decision to separate, Doris Rizkallah took the children -- Suzanne, 7, Donna, 9, and Deanna, 10 -- to live for a month in a friend's basement. She was asked to leave. "You know how kids can get rowdy and everything, and they just said, 'That's it.' "

In the meantime, she obtained a permanent position, making $230 weekly in her company's mailroom. But hunting for an apartment affordable on that salary proved fruitless, and the family turned to Stepping Stones.

"I just found a lot of dead ends," said Rizkallah, who grew up in Montgomery County and graduated from Eastern High School in Kensington. "They'd say, 'You'll need a three-bedroom,' and there are really none in my range."

Now into her second month at the shelter, Rizkallah is still looking for housing, and getting nowhere. The children, embarrassed at the truth, tell classmates that they are living with their grandmother.

"I saw a two-bedroom for $510 a month," she said. "If I bring home $800 and pay out $500 and the rest of it goes for baby-sitting and car payments, and I need a car to get to work and get the kids to school . . . . I thought about getting a part-time job on weekends, but it just didn't make sense because of baby-sitting costs."

Meanwhile, Rizkallah said, she is still facing a bill for back rent of more than $3,000, plus a $212 electric bill. When she tries to repay those debts, she said, "My paycheck is gone before I get it." She toyed with the idea of filing for bankruptcy, she said, until a lawyer told her that it would cost $400.

"I don't know how I'm ever going to get an apartment because we have bad credit," she said. "I don't know if anyone will even consider renting to me."

The best solution, she said, would be to share a house with another family. "But you rarely see an ad in the paper for a ready-made family to share with somebody else," she said. "

Rizkallah's house-hunting problems are far from unique. Greentree Shelter in Bethesda has found that only 40 percent of its residents are able to obtain permanent housing upon leaving the shelter, "and we considered that a fantastic statistic," shelter director Lienhardt said.

Still others are unable even to enter Greentree in the first place.

"I've talked to people on the phone and we've been full, and I don't know what else to tell them," Lienhardt said.

What happens to such people?

"I really couldn't tell you what they do," Lienhardt said. "Generally they don't have 20 cents to call us back and let us know."