Sheila Patrick left her apartment in a Northeast Washington public housing project after a fire last July. Debra Harvey left her apartment in Northwest after the building was converted to a condominium. Gretchen Moore said she left home after a family argument. Parthenia Hall said she moved out of her place in Northeast because of rats.

Yesterday the young women sat eating turkey stew and rice in a bustling dining room at the Capitol City Inn, 1850 New York Ave. NE, one of the city's emergency shelters for homeless families. Over the sounds of children playing and forks scraping on plastic plates, they traded stories about their fruitless efforts to find housing in Washington.

They are low-income families in limbo, caught sometimes for months in a dreary routine of motel rooms, cafeteria-style meals, frustrating trips to real estate offices, and anxieties about the future.

They are among a record high 642 persons from 226 families now in family shelters here, a dramatic increase in recent months that has prompted the D.C. Department of Human Services to launch a new effort to assist such families.

"I am determined that we will find these people housing," said Audrey Rowe, the commissioner of social services, who said the current number is triple that of September 1981 when 209 persons were in family shelters. Rowe said she believes much of the increase is caused by the deteriorating economy.

Rowe said yesterday she is forming a 10-person task force that will be assigned to the shelters beginning Monday to work with families, real estate agents, landlords, city housing officials and community groups to find suitable housing. DHS will then have 16 workers providing housing assistance.

The family shelter program, at the 200-room Capitol City and the 47-room Pitts Motor Hotel at 1451 Belmont St. NW, cost the city nearly $400,000 in September alone.

The city pays $34 to $40 per room each day, plus $10 to $12 daily per person for cafeteria-style food. The shelter program far outstripped its $1 million budget for the fiscal year that ended Sept. 30, although DHS has not yet totaled the amount.

"This cost can't be tolerated, and this isn't even winter yet, when we get all the people burned out and the people with heating problems," Rowe said.

Rowe said economic problems are apparently increasing evictions, which land many families in shelter. Moreover, she said, economic pressures and joblessness are apparently adding to family tensions that frequently culminate in arguments and violence that lead to the breakup of many of the other households.

The program was intended to provide 20 days' temporary shelter, but the current average stay is 41 days. Once families land in shelters, they are often caught in a squeeze between the lack of available subsidized public housing, which currently has a waiting list of 9,000 families, and the lack of affordable housing in the private sector. Most of the homeless families receive public assistance.

In cases where families can't find rentals, Rowe said a major thrust of the new DHS effort will be aimed at "retying the family network" of the displaced. With the aid of social workers, she said, homeless mothers and children might be able to move in with relatives, even in those cases where past disputes led to the breakup.

"In many cases, they may go back into an overcrowded home" temporarily, Rowe acknowledged. "But I would rather have them raise a child in an overcrowded home situation than raising a kid in a hotel. Raising a kid in a hotel seems to me to be the worst alternative."

Rowe said she believes family shelters should be reserved for the destitute who lack any other alternative housing. She said some of the families who end up in shelters probably could have avoided long-term stays if family problems had been resolved quickly after the initial separation.

At the Capitol City, Harvey, 28, said she had been unable for three months to find an apartment for herself and her son Eugene, 11. "I've been checking all kinds of places," she said. Many landlords "won't take PA public assistance people, and the ones that do take PA, I can't afford the rent."

Harvey, who receives $236 a month in welfare payments, said she recently located an apartment for $133, but said the landlord told her he would not accept welfare clients.