It is, and if you choose the number uniformly at random, it's not just "effectively" zero, it is precisely zero.

GP's point, as I understand it, is that it is not actually possible to choose a number from [0, 1] uniformly at random in "real life".

I think you could argue that, e.g. in the dartboard thought experiment, the probability of choosing individual points doesn't really matter: only probabilities of measurable subsets with positive measure matter.