You can't pick a random real number between 0 and 1. Heck, almost all reals between 0 and 1 can't ever be constructed let alone picked.

The here is the non-constructive nature of the real numbers. That is not to say the reals are useless, but they are not much more than a formalism. It's rather useful though because it's hard to get numbers like pi or e. Its really nice that any real interval is compact, but that too is hard to replicate.

Non-countable sets defy intuition on several ways. The silver lining is that we don't have any evidence a non-countable thing exist on the real world.

I don't think anybody even has a procedure for gathering that kind of evidence.

It is possible! Actually I remember​ pointing out a similar concern in my probability class back in the day. The teacher's answer: that is precisely the difference between probablity and possibility :)

I would argue that the distribution you used to pick a random number was not uniform. Not all the real number were equaliy likely to be picked by you. Hence the probability for some numbers was > 0.

Is it probability actually zero, or just infinitely close to zero?