Electronics, CPUs, etc. The drill not so much.

Oh, you mean the payload. I'm not sure but I don't see why we couldn't quarantine that for a while and sit on it. I take it we've gotten other space rocks including meteorites and moon rocks. They haven't caused us much trouble so I don't see why an asteroid would be worse on that front. Then again, I'm no astrophysicist.

Radiation does not work the way you think it does.

Just about everything is slightly radioactive so saying something is radioactive is next to meaningless, it's a question of scale with some things being 1,000,000,000 times as radioactive as other things even though they are both 'safe'. The surface of an asteroid is probably be slightly more radioactive than average but they are planing on minding 100-500meter wide objects the vast majority of which are shielded by the ice and rock above them.

PS: Technically even 'pure' vacuum is slightly radioactive.

Ah, there's ways of dealing with that. Copper mines in Australia, for instance, often have trace uranium they have to deal with before they can sell it, and there are ways of dealing with this. It's easy to separate different elements, especially given the difference in density, and though it's usually harder to separate isotopes (U-238 and U-235 being the famous example) it really depends on the metal you're talking about. In addition, enriching uranium is a problem because nobody wants to help you do it, but with most other metals, everyone's cool, and would be happy to sell you that service.

Keep in mind not all radioactive sources are created equal--every atom is different, and every isotope is different. Iron and nickel, for instance, are extremely stable and can be irradiated without serious consequences--unlike cobalt or gold, which can be added to a bomb to make it dirty.

So, because the bulk of the asteroid is very stable, and the remainder of it very precious, I would expect dealing with the radiation not to be a problem. The precious metals may in fact be somewhat irradiated, but then this is not waste--radioactive isotopes of many different metals see a lot of use in industry and medicine.