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.