Yes, it's uncountable, so the probability of any particular real number is zero. I'm not going to go into detail by what I mean by a probability distribution here (though I'll note that cousin it used the word "integrate"). To uniformly pick a number $x$ from 0 to 1, use an infinite sequence of coin flips to represent it in binary. Infinity is impractical, but for natural questions, like: is $x$ between 1/5 and 1/sqrt(2), you will almost surely need only finitely many flips (and your expected number is also finite). And the probability that the answer is yes is 1/sqrt(2)-1/5; that is the sense in which it is uniform.

You can transfer this distribution to the set of all real numbers, eg, by logit or arctan, but it won't be uniform in the same sense. One can satisfy that uniformity by a measure which not a probability measure.