Their price range was around $2,000 a month, preferably less. In Brooklyn, one-bedrooms near the Navy Yard were too pricey, so they looked south of Prospect Park. One place was plagued by an incessant hum from the boiler room, “like being in the belly of a ship,” Ms. Donnelly said. “It would drive us insane.”

They saw top-floor apartments with many steep stairs and ground-floor apartments with air-shaft views. Many had walls thick with layers of drippy paint. In some cases, Mr. Routh said, the feeling was, “I guess we could live here, but let’s keep looking.”

Some one-bedrooms were so tiny they seemed suitable for only one person or, better still, “for an introvert who never had anyone over,” Ms. Donnelly said. “We didn’t want to be limited in the size of a party we could have.”

Image A one-bedroom for $1,900 in a building on Edgecombe Avenue in Harlem “barely fit the bed and a small nightstand,” Mr. Routh said. Credit... Katherine Marks for The New York Times

On Edgecombe Avenue in Harlem, an apartment renting for $1,900 seemed to be part of a larger apartment split in two. “We brought our tape measures along, and it barely fit the bed and a small nightstand,” said Mr. Routh, who sketched out the furniture on graph paper.

Up in Hamilton Heights, a two-bedroom for $2,000 had a living room that could hold little more than a sofa. And it overlooked 145th Street, the crosstown emergency route, which was loud and busy — bad enough on a Saturday morning and no doubt worse during rush hour — with a gas station across the street.

Through Naked Apartments, the couple met Rachel Zirkle Miller, then a licensed saleswoman at Bohemia Realty Group, which specializes in Upper Manhattan. She took the couple to see a one-bedroom renting for $1,825 on a tree-lined block, a short walk from 145th Street.