Please describe how it is possible to pick such a number. For example, I can readily imagine how to pick a random 32b float, but that it is an entirely problem with a nonzero probability.
In probability theory, when dealing with continuous sample spaces / random variables, events with probability 0 still have a chance of occurring, and events with probability 1 stil l have a chance of NOT occurring, see:
In terms of implementation, I'm not aware of an algorithm that can randomly pick a real number on an actual computer. Perhaps a mathematician could show how to pick one on some abstract machine with infinite resources, and not constrained by finite bit representations of numbers.
A Turing machine can't pick random numbers of any kind.
Once you accept that you have an entropy source in the physical world, you can easily be injecting random real numbers (from some range) and in fact, usually are, which are then being binned into integers by ADCs.
https://en.wikipedia.org/wiki/Almost_surely
This strange property comes from strange properties of the real numbers (and uncountably infinite sets) that give rise to things like:
https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
Measure theory deals with resolving this:
https://en.wikipedia.org/wiki/Measure_(mathematics)