I'm looking for a short story
"The Good Work" by Theodore L. Thomas; first published in If, February 1959, which is available at the Internet Archive.
that takes place in a sort of utopian future, where no one has to work because machines do everything. People can basically sit and do nothing all day, but this is unsatisfying for some.
"I don't know. Some people work; there must be jobs somewhere. I suppose they get them through the Ministry of Government Employment, and you know what people say about that. Government workers won't even talk about it; everybody says they're ashamed of it. I don't know what's going to happen. Except—I'm through. I'm going to take my stock home with me tonight, and that ends it."
They end up getting a job where they go around and tighten screws
"Yes, Mr. Winthrop. We have a job for you, and the full six hours a day, too. You will be on the maintenance crew of your building. Your job is explained here—" he passed over a card—"and it consists of tightening the nuts on the expansion joints in the framework of the building. It is very important to do it right, so read the card carefully." Winthrop nodded eagerly.
that are loosened on the machinery due to their vibrations.
The chief of the tightening crew was a big, bluff man with a red face. He took Winthrop in tow and showed him how they worked. The crew chief had a vast knowledge of the crawl spaces in the interior of the building. He showed Winthrop the blueprints from which the tightening crew worked, and explained that by coordinating their work with all the other tightening crews they made one complete round of the building every eight years. By then it was time to do it again; the nuts worked loose from the constant expansion and contraction. It was quite a job keeping track of the area that the tightening crew covered; it was a large crew. But each member turned in daily reports, and there was a large clerical staff to keep the records straight. In fact, there were more men keeping records than there were doing the actual tightening work. The chief pointed out that Winthrop was to be one of the elite, one of those whose work justified the existence of the huge staff. The tone of the chief's voice made it clear that there was a kind of quiet pride among the men who did the actual work. The chief issued Winthrop his wrench and showed him where to start.
At the end of the story, we learn that another crew wakes up every night and goes around loosening screws.
Winthrop walked in and the two men stood looking at each other. Winthrop was surprised at how well Barlow looked, and he said so.
Barlow laughed. "Yes, the last time we met I was pretty far down in the dumps. But I'm working, Jeremiah. I'm actually working. Important work, too!"
His enthusiasm was infectious and Winthrop found himself laughing. "I'm glad for you, John. And I know how you feel, because I'm working too."
[. . .]
There was an exchange of goodbyes, and Winthrop left.
Barlow went into the other room and came out immediately with his wrench. He waved it playfully at his wife. "Got to go," he said. "The loosening crew won't wait." And he blew a kiss at his wife and went off to work.