As I understand it (see: not at all), if you leave a spaceship with no suit on, you’d get baked like Marie Curie’s ovaries from the radiation. It’s mainly our atmosphere that protects us from most of the nastiest stuff. Would a giant cable reaching from Earth all the way to a platform outside the atmosphere become dangerously-radioactive over time? And if so, would that eventually cause the entire planet to get radioactive over hundreds of years? Kinda like if the hole in the Ozone layer were replaced with a Mario pipe.
And if that is the case, maybe we could forget the elevator aspect of it and just aim for a free eternal source of radioactive energy, like a really shitty Dyson sphere 👀
Ada answered the question, but consider your question.
You said “leaving a spaceship with no suit”. If your idea is that something left in space becomes dangerously radioactive then any space station or space ship would itself become dangerously radioactive.
So the answer is, no, things in space don’t become dangerously radioactive. Also things in contact with radioactive substances don’t themselves become radioactive except under extremely specific circumstances. Your house didn’t become dangerously radioactive because of the radioactive americium-241 in the smoke detectors.