In every power plant there will be a risk of injury. High energy steam, regardless of what boiled the water, is a hazard, as well as all the electrical equipment, chemical and atmospheric conditions.
The Navy takes great care to keep the engine room safe, for both personnel and equipment. Every time someone gets injured, an investigation is launched to find out what happened, how it can be prevented in the future, and spread the word to everyone else in the fleet about the findings.
For example, the administrative requirements to prove that a certain piece of equipment is electrically de-energized prior to getting permission to perform maintenance on that equipment is a nightmare, but it's written in the blood of sailors who've been hurt. I took me a long time to accept that it would take over 4 hours of administrative headache to get to the point where I could perform a 10 minute maintenance item. But this is the way it has to be, to keep people safe, and keep the equipment safe.
Legally, any exposure to ionizing radiation invovles some risk of cancer. Out of 20 years in the Navy, being on 4 submarines and 1 land-based prototype reactor plant, I have just over 1.5 Rem of lifetime exposure. Of all the people in a group of 10,000 who get cancer, I might be 1 more. Might. But what if it was due to exposure to cosmic radiation while mountain climbing? There's no way to know, just analysis of the statistics. There are even health studies on the internet showing that cancer rates among nuclear workers are actually lower than that of the general population. The average exposure to people living in the US is 355 mRem a year, most of it due to Radon. While on board the USS Chicago, I averaged about 8 mRem a month. It is ironic that now, while walking around Portland, Oregon, I get almost 4 times the radiation exposure I received while working 50 feet away from a reactor.
In the course of my 20 years, I have seen only 1 injury that required us to Medivac someone off the boat. A mechanic could feel a small amount of air coming out of a low-pressure air compressor. A flashlight inspection revealed nothing, so he decided, on his own, to stick his arm deep into the machine to try to feel the source of the air. Unfortunately, it was from a fan, and he proceeded to stick his hand into the rotating blades. The corpsman on board stiched his fingers back together, and we headed for the nearest place to meet up with a helicopter to get him to a hospital. He came out OK, but let his curiosity over-ride his following procedure: turn it off before you go poking around!
I have seen 2 or 3 people take an electric shock, but the Navy threshold of what voltage is considered dangerous is only 30 volts (yes, thirty). Safety training is one of those things that gets hammered into you over, and over, and over....the other postings are correct, you're most likely to get a paper cut and look for a cooler spot than get injured or get cancer from the radiation.