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.

You can certainly pick a random real number from the unit interval.

Really? Go on, then, pick one and tell us what it is (or at least tell us what your procedure was).

One example in a finite space and time setting would be selecting a random point on the ground. Say by dropping a ball there or something. The exact coordinates it lands is a random real number. But the probability that it landed on those exact coordinates is exactly 0, hence a paradox.

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.

No. Rational number are countable, but have the same property of zero probability for an item in a uniform distribution.

Why the downvotes? I think he is correct; and if someone think he is not, please elaborate

Yes, no reason for the downvotes.

Infinities are problematic too.

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?

If it's a random real then the probability is zero. If it's an arbitrarily close approximation to a random real then the probability is arbitrarily close to zero.

How can the sum of infinitely many zero probabilities be 1? I can understand how the sum of infinitely many infinitely close to zero values can be 1, but not infinitely many exactly zero values

There's no such thing as a "sum of infinitely many" anything. What we are talking about is the limit of an infinite series, which behaves nothing at all like a sum.

Those are the same thing. You're just calling them by different names.