I'm not sure either. It's not an air tight bomb shelter and it's used like an average storeroom, storing stuff like winter jackets, suitcases and paint. I do use small amounts of Calcium Chloride based dessicants to keep the room dry.
Radon is a noble gas and chemically inert. It's radioactive, which is why it's a toxicity worry. But there was enough radiation in that basement to mechanically damage plastic in three years, OP would be long since dead.
Radon is a noble gas. It's not going to affect fiber or wiring... but yes, if you're going to spend much time down there, definitely a good idea to check for it.
Interseting - I have high radon (mitigeted in the house via a fan), if I burry fibre what will I need to do to get something that will work? I have an outbuilding that I want to give internet.
Well technically a nuclear bomb would also degrade the jacket of fiber cabling pretty badly, but we don't really concern ourselves with that since it means you're dead and the house is gone anyways.