It was. The Reagan is nuclear-powered and is capable of carrying nuclear missiles (and quite possibly was at the time). Unless Navy planning has entirely rotted away, that ship was plastered with radiation detectors, crewed with people trained for nuke decon, equipped with NBC (nuclear/bioweapon/chemical weapon) protection suits, and supplied with potassium iodide tablets for thyroid protection.
I am a CBRN specialist in the Marine Corps. These ships run constant radiation detection, everywhere, especially in the water desal. The only probable cause for these kind of biological responses, would have to be inhalation of radiated particulates, but even that would be extremely difficult given the range from the source. Even the most basic respiratory protection(shirt over face) would prevent most of that type of contamination.
The entire CBRN community was hyper focused on these operations Navy wide, and the possibility of an entire group of ships not taking every precaution is very very very low. I am not discounting the possibility of equipment failure(unknowingly) or execution failure by individuals, but the chances of that with so much attention on the issue would be low.
EDIT: What do people do when they see snow? Think tounge out.
Probably mass hysteria. It seems unlikely that the captain blithley ignored basic safety measures, and the symptoms reported in the article are not typical for radiation sickness.
