A job transfer of sorts brought Katie Muela and Arsy Khodabandelou to New York from Maryland a few years ago.

Ms. Muela was working as a nanny for a family that was relocating to the city from the Washington, D.C., area, so she and Mr. Khodabandelou took it “as an opportunity to move to New York for fun,” she said.

That was just before Covid shut down the city. Disappointed, the couple retreated to Maryland after a year. But after the pandemic, they were eager to return, and landed in a one-bedroom on the Upper East Side for around $2,600 a month.

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They fell in love with their bustling Yorkville neighborhood. “We got to know our neighbors and see them at the dog park every day,” said Ms. Muela, 33.

One day, the two went hiking at Bear Mountain, about an hour and a half north. Mr. Khodabandelou, who works in software sales, found himself worrying about their dogs, who were home alone. Friends and family were back in Maryland. It gave him the idea to hire a software engineer and create a pet-rescue app, Woof, which notifies emergency contacts if pets are left alone longer than expected.

The nascent business landed Mr. Khodabandelou, 35, on the TV show “Shark Tank” in 2023, though it didn’t land him a deal. Nevertheless, he said, “It was the coolest experience of my life.”

Meanwhile, the couple, who will marry next month, were outgrowing their small apartment. In the basement laundry room, the machines were often broken. And they had little counter space, making it difficult to cook and host guests.

“We didn’t have anywhere to sit,” said Ms. Muela, who now works as a court reporter. “We ate on our couch. Why, at this age and stage of us wanting to start our future, would we not purchase?”

Some of Mr. Khodabandelou’s old friends were having children and buying houses. “So it was in the back of my mind,” he said. “Do I want to be in New York or go back to Maryland and buy a house?”

The couple decided to stick around and get an apartment — dog-friendly, of course — near their rental, on the far Upper East Side. They aimed for a two-bedroom or a large one-bedroom, and quickly realized that they’d have to adjust their budget.

“We thought $500,000 was our max, but we couldn’t get anything for $500,000,” Mr. Khodabandelou said. “We kept raising our price.” It ended up at $800,000.

Online last winter, they found Annette Mulligan, then a licensed salesperson at Keller Williams NYC and now at the Corcoran Group, who explained the difference between co-ops and condominiums and noted that dog policies differ among buildings.

“I am not a picky person,” Mr. Khodabandelou said. “I fell in love with every single place that we saw.” But those places often sold quickly.

Ms. Muela was more particular. “I definitely was a brat at first,” she said. “I wanted two bedrooms, a washer-dryer and outdoor space.”

As time passed, she lowered her expectations. Among their options:

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