Nubia Chavez and Manuel Acuña have been bouncing from rental to rental around New York City like millennials. They recently decamped to a $700-a-month room in a Queens apartment, with a shared bathroom and no access to the kitchen except to make coffee.

But Ms. Chavez, a housekeeper, and Mr. Acuña, a retired building porter, are not in their 20s. They are 65 and 72, and they say they are tired — of the moving, of the lack of permanency and of a lifestyle not suited to their age.

“I want to live in my own place,” Ms. Chavez said. “No more rooms.”

Jennifer Stock, 33, has been looking for a place for her ailing 89-year-old father after his assisted living residence in Park Slope, Brooklyn, announced it would shut down by this summer.

And in Astoria, Queens, Norma and Rodolfo de la Rosa recently put their names on a waiting list for an affordable residence for older adults because, they say, their Social Security checks cannot keep up with rent increases. The list has nearly 4,000 names.