Reactors on Navy vessels, since at least the 70s, are very safe. They are closed systems and are completely sealed in compartments surrounded by feet of shielding. While some negligible amount of radiation does get through, it's half life is very short. Radioactive isotopes don't escape, nor do they get created by proximity to a radiation source and start floating around the crew compartment. In addition, each crew member carries a dosimeter which tracks their exposure. As others pointed out, walking around outside exposes us to more radiation (from the sun, earth, ceramic tiles...) than spending months next to to one of these reactors.
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