Yes. They tried to send in robots to observe and help the liquidators move debris at Chernobyl, but they didn't last very long. It isn't so much the radio signal as the electronics get damaged very easily by radiation.
It doesn't effect radio communications. It does very nasty things to electronics though. Mostly because it can provide energy directly to the inside of, say, a transistor or the inside of a dram cell causing things to malfunction. It can also do interesting things to the silicon lattice itself, which is what typically causes complete failure of electronic devices in a high radiation environment.
As far as I can recall, most data cables arn't shielded. Audio cables usually are, however, to stop hum/etc. when travelling over long distances.
To be more specific, if you have wire it will act like an antenna. It's usually a fairly poor antenna for most types of EM that you would find, however there are some things, ie. 50hz hum from mains power and its harmonics, that are powerful enough to induce a nontrivial amount of interference for analog applications.
Most digital signals either don't care (because the interference is too weak), or don't care (because they are using twisted pairs and the signal cancels out overall) and hence they are usually unsheilded.
Nuclear radiation can't, in normal circumstance, induce a signal into a wire.
