The story was first published in Astounding Science Fiction in 1940, before any nuclear reactors had ever been built, and later reprints of the story required some modifications to reflect how a reactor actually worked.
The story is one of the earliest in Heinlein's Future History chronology, taking place in the late 20th century. But for another story "Life-Line", which is not particularly relevant to the Future History, it might actually be the earliest.
It describes the tensions amongst the staff of a nuclear reactor. Heinlein's concept of a nuclear reactor was one of a barely contained explosion, not the thermal piles developed later. As a consequence the work is dangerous, and the slightest mistake could be catastrophic. All the technical staff are monitored by psychologists who have the authority to remove them from the work at any time lest they crack under the pressure and precipitate a disaster. Needless to say, the monitoring itself is part of the problem.
Using a method called "calculus of statement", some theorists come to believe that calculations on the stability of the reactor have greatly underestimated the scale of the reaction should the reactor go out of control. The situation seems hopeless, as the energy produced by the reactor is sorely needed on Earth, oil having been monopolized by the military.
However, there is a way out. One of the by-products of the reactor is a more stable nuclear fuel which can also be used as the basis for a rocket engine. Armed with their theories and the new fuel, the protagonists undertake a campaign to have the reactor shut down, moved into space, and used as a source for the fuel, which will supply the needs of Earth and take humanity into space. Their final card is a shame campaign which will subject the trustees of the reactor to public vilification.
The next story sequentially is "The Man Who Sold the Moon". In that we find that the reactor exploded in space. The actual cause was the detonation of the service rocket's fuel, caused by the effects of cosmic radiation on the supposedly stable nuclear material.