this post was submitted on 26 Nov 2023
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The radiation concerns come from the radium-based luminescent paint used on watch dials prior to the early 1960s. The watch in your link does not actually appear to have any luminescent paint.
Having said that, the majority of radium decay is alpha decay, which can be blocked by a sheet of paper, though a very small fraction is beta decay (blocked by a sheet/plate of aluminum) and gamma decay (blocked by a lead vest, much like what you would wear for an x-ray, or a couple inches of concrete). As the half-life of radium-226 is 1600 years even watches that do not glow in the dark anymore remain radioactive if they used radium lume when they were first produced.
The initial form of radium decay is to radon (yes, the radioactive gas) but it's minimal enough that a well-ventilated space is sufficient to disperse the radon (because radon's primary health effect on the human body is lung cancer, it would be years spent in a poorly-ventilated or unventilated space with a radium watch before the radon buildup induced lung cancer).
Wow, excellent reply. Thank you for the knowledge.