I'm not sure if this is entirely mathematically valid, but it's at least intuitively valid. Imagine we're in decimal (for familiarity). Imagine we're generating our random number by drawing digits out of a bag containing the ten digits, then replacing and drawing again. This is of course a magical perfectly uniformly distributed bag.

In order to draw a rational number, you must randomly select a number that has a repeating pattern in decimal. Imagine that for the sake of argument that we're in this "repeating pattern" and, amazingly, we've already drawn it 1000 times! What a roll we're on! In order for the number to be rational, how many more times must we draw this pattern? Infinitely many. What is the probability that in one of those infinite fair random draws, the repeating pattern gets broken? 1. It's just not possible for infinitely many fair draws to produce a repeating pattern; that would be proof that the process generating the number was in fact not random in the first place. (I can't be that sloppy if we're doing finitely many draws but I believe that's justifiable at infinity.)

Imagining through this process of drawing a random number I think makes this more intuitively obvious than imagining being presented with a completed number and trying to figure out if it's rational.

The measure of epsilon is 0.

Proof: what else could it be? If it's not 0, there's a smaller number, contradicting your (intuitive) definition of epsilon.

Epsilon is a variable, not a number, so that wouldn't make sense. The probability is less than epsilon for all epsilon > 0. The only non-negative real (and we know the probability must be some non-negative real) satisfying that is 0.