I got this one without realizing the complex basis for it... there's an infinite number of real numbers between 0 and 1, compared with a limited proportion of rational numbers.
A way to intuit why you have a zero probability of choosing a rational is to consider randomly picking digits one at a time. Suppose you start with a "3". What is necessary to get one third as your random choice? You must now uniformly randomly select an infinite number of 3s. You can think of that as just a contradiction without much loss here, in that selecting an infinite number of 3s is infinite proof that your process must not be random.
(Putting rigor on that might be a challenge but I think it's a fine intuition.)
This argument similarly holds for any rational, because no matter how large any given rational's repeat may be, it is always finite, and in order for that rational to be your choice you have still fully determined the result of a putatively uniformly random process for an infinite number of exact picks, and if it is truly random, the probability of that process producing the exact necessary repeat infinitely is always 0.
This is almost exactly the same thing as saying integrating x^3 from minus infinity to plus infinity is zero because each zero width line has zero area, so adding them all up is zero too.
Gives the right answer by coincidence, but also works to show incorrect results.
In fact, the probability you pick an uncomputable number is 1, too.
One way of interpreting this result is to say that randomly selecting a real number isn't physically meaningful and this does not fit into our human brains well. You can do math on the "random number", but you can't ever actually have one in hand (as a sibling comment to yours points out correctly, it is guaranteed to be uncomputable as well), not even mathematically, so you should expect counterintuitive results. Which is generally true of the real numbers when you get close enough to them anyhow. They are popular for a reason but they have rather more spiky bits and strange behaviors than one might expect from a number called "real".
I think both of you may be thinking of "random number" as something like what you get out of a RNG or something, but random reals are much stranger than random ints from a finite range.
("You'll never land on Pi because the odds of picking out the infinitely many digits is zero"- repeat that for every irrational, and naively sum that up, and it looks like you'll never land on an irrational either, which is not true.)
It does not.
[0] https://en.wikipedia.org/wiki/Cantor's_diagonal_argument