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This story featured a young couple who were looking for an apartment. They didn’t have a lot of money to spend and were very excited to find the perfect place, all the amenities, unbelievably low rent, well located, everything. The other renters were also young, and everyone was thrilled to have found such an unbelievable deal.
One day when everyone was at home the apartment closed up, everyone was trapped, and the entire building lifted off into space. It was a trap, designed to capture humans. I think the couple escapes in the nick of time.

"Shipshape Home", a short story by Richard Matheson, also the answer to this old question and this one; first published in Galaxy Science Fiction, July 1952, available at the Internet Archive.

This story featured a young couple who were looking for an apartment. They didn’t have a lot of money to spend and were very excited to find the perfect place, all the amenities, unbelievably low rent, well located, everything.

"No, it's fluky," Ruth analyzed as we sat having dinner that night.

I grinned at Phil and he grinned back.

"I think so, too," Marge agreed. "Whoever heard of charging only sixty-five a month for a five-room apartment, furnished? Stove, refrigerator, washer—it's fantastic!"

"Girls," I said, "lets not quibble. Let's take advantage."

"Oh!" Ruth tossed her pretty blonde head. "If a man said, 'Here's a million dollars for you, old man,' you'd probably take it."

"I most definitely would take it," I admitted. "I would then run like hell."

"You're naive," she said. "You think everybody is Santa Claus."

"It is a little funny," Phil said, "think about it, Rick."

I thought about it. A five-room apartment, brand-new, furnished in even better than good taste right down to a couple of sets of expensive dishes . . .

I pursed my lips. A guy can get lost writing about the bars on Mars. Maybe it was true. I could see their point. Of course, I wouldn't show it, though. And spoil Ruth's and my little game of war? Never.

"I think they charge too much," I said.

"Oh, Lord!" Ruth was taking it straight, as she usually did. "Too much? Five rooms yet! Furniture, dishes, linens, a television set! What do you want, a swimming pool?"

"A small one would be good enough."

She looked at Marge and Phil. "Let us discuss this thing quietly. Let us pretend that the fourth voice we hear is nothing but the wind in the eaves."

"I am the wind in the eaves," I said.

"Listen," Ruth restated her forebodings, "what if this place were a fluke? I mean, what if they just want people here for a coverup? That would explain the rent. You remember the rush on this place when they started renting?"

I remembered as well as Phil and Ruth and Marge. The only reason we'd got apartments was we all happened to be walking past the place when the janitor put out the renting sign. The four of us had gone right in. I remember our amazement, our delight, at the rental. Why, we'd been paying more than double for half the size and ratty furniture, besides.

One day when everyone was at home the apartment closed up, everyone was trapped, and the entire building lifted off into space. It was a trap, designed to capture humans.

I headed for the door and grabbed the knob.

It didn't turn.

A bout of panic drove through me. I grabbed at it and yanked hard. I thought for a second, fighting down fear, that it was locked on the inside. I checked.

It was locked on the outside.

Marge was ready to scream. You could sense it bubbling up in her.

"It's true," Ruth said, horrified. "Oh, my God, it's true then!"

I made a dash for the window.

The place began to vibrate, as if we were about to get hit by an earthquake. Dishes started to rattle and fall off their shelves. We heard a chair crash onto its side in the kitchen.

"What is it?" Marge cried.

Phil grabbed for her as she began to whimper. Ruth ran to me and we stood there, frozen, feeling the floor shake under our feet.

"The engines!" Ruth suddenly screamed. "They're going now!"

"They have to warm up!" was my wild guess. "We can still get out!"

I let go of Ruth and grabbed a chair. For some reason I felt that the windows would be automatically locked, too.

I hurled the chair through the glass. The vibrations were getting worse.

"Quick!" I shouted over the noise. "Out on the fire escape! Maybe we can make it!"

I think the couple escapes in the nick of time.

Spoiler:

And then the sky, which was growing light, grew dark. My head snapped around. Women were screaming their lungs out in terror. I looked in all directions.

Solid walls were blotting out the sky.

"We can't get out," she said. It's the whole block."

And then the rockets started.