The paradox is that, after picking a random number, you have just done a thing which has probability zero. Doing a thing that has zero probability shouldn't be possible. Ever.
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.
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.
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.
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.
