About 270 runaways and homeless youths at one time or another last year found their way to a dingy, off-white, stone and brick turreted house not far from Dupont Circle.

There, a local nonprofit organization called Special Approaches to Juvenile Assistance provides temporary shelter for the kids, whose ages range from 11 to 18, while the staff tries to reunite them with their families or find alternate placement for them. "If this program did not exist, these kids would be out on the streets--and various other places," says Cornelius Williams, SAJA's executive director.

The program last year was supported by a $68,000 grant from the Department of Health and Human Services. This year, it received $10,000.

The group's problem--to compensate for 58,000 fewer federal dollars--is but one example of what a wide assortment of private, nonprofit organizations all over the country are facing as the Reagan administration's budget cuts begin to have an impact.

Compounding the problem nationally is the fact that changes in tax laws may actually work against increased giving by both individuals and foundations, according to those in the business of philanthropy.

A change in the law also allows foundations to disburse less each year in the future than they were required to disburse annually in the past.

No one really knows what will happen. Local officials are just beginning to grapple with what promises to be a mammoth problem: trying to distribute the fewer federal dollars they will get through block grants among increasing numbers of seekers whose alternate federal sources have been cut. Because of cuts in federal services, there will be increased demand on the nonprofits to provide those services--but they aren't likely to have the wherewithal.

Volunteerism and increased charitable contributions are supposed to take up the slack, but it's hard to find anyone involved who thinks they will.

"There is a major concern on the part of foundations and the national corporate world that something must be done," says Lawrence S. Stinchcomb, president of the Community Foundation of Greater Washington, "but also there is the realization that there is no way for the foundations and the national corporate world to fill a $45 billion gap. The arithmetic just doesn't work."

A gap of $45 billion is the amount some estimated last year would be lost by nonprofit organizations between 1981 and 1984 as a result of the combined impact of budget and tax changes proposed by the Reagan administration.

"If that's the gap created by federal cuts, we're talking about such large numbers that private-sector philanthropy can't possibly fill it," Stinchcomb said. By comparison, he noted, the corporate world's donations total about $2.5 billion nationally each year, and the foundations contribute about the same amount.

No one seems quite sure of the impact, since the budget process continues. The Urban Institute study estimated that, based on President Reagan's March 1981 proposals, nonprofit revenues from federal sources would decline by $27 billion over the next four years as a result of the budget changes, and by a further $18 billion as a result of tax changes.

The negative impact on individual charitable giving outlined by the study was based on analyses of changes in tax rates, changes in income levels that affect tax brackets, decisions to itemize and the resources available for charitable contributions.

Estimates of the potential shortfall locally are hard to come by, but James Kalish, executive director of Washington Council of Agencies, an association of area nonprofits, estimates it could be as high as $80 million a year.

Rev. Raymond Hartzell, executive director of the Lutheran Social Service Agencies, says there has been a slight increase in volunteerism, but nothing significant. "The needs are so great that there would have to be a vast outpouring of people's money and time to make up for the cutoffs and cutbacks," he said. "It would have to be an avalanche."

The economic times don't instill optimism for Hartzell's "avalanche" of private giving to compensate for the missing dollars. Contributions to charities, including United Way, have barely kept up with inflation over the last few years. Some local nonprofits already have seen a reduction in the aid they used to get from the community, and they seem somewhat understanding.

"The business community has always been pretty supportive of us," says Ron Clark, executive director of RAP, a drug rehabilitation program. "But because of the recession, they have to pull in their belts and can't pass on to us the contributions they used to. It seems to be a chain reaction."

Contracts totaling almost $180,000 will see RAP, which provides 24-hour residential and support services to as many as 58 young men and women at a time, through mid-year, Clark says. But he has seen a decline of in-kind services they also need, donations ranging from bread to two-by-fours.

One way in which some nonprofit organizations are trying to cope with the prospect, and reality, of reduced funds is to develop some income-producing activities. For instance, the Philip L. Graham Foundation recently made a $10,000 grant to the Latin-American Youth Center to develop an income-generating plan. The center is now training young Latinos for jobs in auto mechanics, carpentry, catering, office management and silkscreen and graphic design. The idea behind the grant is to see whether the center can generate some income by providing services in the neighborhood through an auto shop and graphics workshop.

The organizations also are beginning to band together somewhat, under the umbrella Washington Council of Agencies, in a cooperative effort. According to director Kalish, the group is trying to determine what the nonprofits can sell to each other that they would otherwise buy commercially--such as computer, mail and printing services.

They may have combined buying power, he says, or combined investing power, if they pool the occasional excess cash they have.