I guess, but, the set is not even countably infinite. "Selecting at random" is something that happens in the real, non-infinite world, not in the mathematically rigorous would where infinities can exist. So, no, probability-zero events do not happen in either.
> Probability-zero events happen all the time. The probability of getting any specific value selected uniformly at random from the unit interval (say, 0.232829) is zero.
I would strongly challenge that claim. First, you did not choose that number uniformly at random, you chose it from at best a countably infinite subset, or more realistically, from a finite subset. And secondly, I do not think you can describe a situation where a number is actually chosen uniformly and randomly from the unity interval.
How?