This is not quite accurate. You can’t uniformly pick a random rational number from 0 to 1, because there are countably many such numbers, and any probability distribution you assign will have to add up to 1. Every probability distribution on this set assigns a nonzero probability to every number.
You can have a uniform distribution on an uncountable set, such as the real numbers between 0 and 1, but since you can’t pick an arbitrary element of an uncountable set in the real world this is theoretical rather than a real-world issue.
As far as I know, any mathematical case in which something with probability 0 can happen does not actually occur in the real world in a way that we can observe.
I very much appreciate this post, because it strongly resonates with my own experience of laziness and willpower. Reading this post feels less like learning something new and more like an important reminder.