For Ms. West, whose youthful appearance belies her age, in her mid-50s, the nights spent on couches in other people’s homes were uncomfortably familiar. She grew up an only child in a housing project in Neptune, N.J., where her mother slept in the lone bedroom, and she occupied a pullout sofa in the living room.

“I’ve always had this dream of doing better,” she said. “I always wanted to own my own house.”

She realized that dream shortly after arriving in Cleveland with her husband and two children in the early 1990s. At first, they rented. But one fall afternoon, Ms. West found herself on a block lined with leafy trees in Mount Pleasant, a neighborhood east of the Cuyahoga River that was a magnet for middle-class black families like hers. Red brick homes with wooden porches sat on ample lots. Public schools were a few blocks away.

When she saw an ad in the Sunday paper offering a house on that very block, she bought it for $45,000; for the $9,000 down payment she used the savings her mother had left her when she died. She and her husband assumed the mortgage from the previous owner, with affordable payments of less than $400 a month.

Image Sheri West in the West Side Catholic Center. By December, she will exhaust its 90-day limit. Credit... David Maxwell for The New York Times

Ms. West then had a job as a maintenance worker at an apartment complex for about $9 an hour. Her husband earned about $10 an hour as a truck driver. As the years passed, they added shrubbery to the front yard and photos of children’s birthday parties to the walls.

“I thought that was going to be my house,” she said.

She tapped her inheritance to buy another house on nearby Union Street, paying $15,000 in cash for a light-blue, vinyl-sided A-frame. She turned the house into a home for five homeless people. She did their laundry, reminded them to take their medications and cooked meals, while collecting payments of up to $750 a person each month from the agencies that placed them.

Over the years, Ms. West and her husband spent more than they earned. They used credit cards to finance restaurant meals. They bought a new S.U.V.