If you pick a real number at random, what's the probability it will be rational?

Not zero. I think the proper way to define it is “almost never” when infinite are involved: https://en.wikipedia.org/wiki/Almost_surely

I’m not sure if that means it will for sure happen or not, if the draw happens an infinite amount of times.

In the standard measure theory formalization of probability, that outcome set does have measure exactly 0.

Perhaps we could use a variation on in which somehow supports some infinitesimal values, but if one wishes to keep both countable additivity and have the distribution be uniform in the sense of translation invariant, then we run into difficulties.

There is an infinite amount of numbers between 0.0 and 1.0, none of them are 2.0