I'm looking to find a short story that takes place on the moon (I believe) and involves workers responsible for taking care of a power plant (nuclear, I think) that powers the entire earth.

Due to its instability, the workers must concentrate intensely while on the job lest the whole thing explode. The stress of constantly living on the edge of disaster leads to men cracking and even attempting to trigger an explosion.

A prominent psychologist (who may be a recurring character by this author?) is brought in to assess the situation but concludes there is nothing that can be done relieve the stress. In the end, however, the power plant is stabilized and the stress removed, saving everyone.

What story is this?

"Blowups Happen" (1940; revised 1946), by Robert A. Heinlein

Your description is a close match, especially regarding the investigation into the plant workers' mental health and stressful job responsibilities. The power plant is on Earth, and the revelation of an ancient disaster involving a similar, alien power plant on the Moon kicks the story into high gear. The story ends with the threat to humanity neutralized as the power plant is shut down in preparation for moving it off-planet.

"Blowups Happen" first appeared in in Astounding Science Fiction (September 1940). Heinlein updated it for its next appearance (The Best of Science Fiction, 1946), but he eventually preferred the earlier version (Expanded Universe, 1980).

You can read "Blowups Happen" online at the Internet Archive.

References:

• 
  Wikipedia has a synopsis.


• The Internet Speculative Fiction Database lists the publications that include this story.