An uniform distribution over the real interval e.g. [0,1] is possible. An algorithm is the fair coin tossing for each binary place 0 or 1. In the case of all tails you have 0. In the case of all heads - it's 1. An uniform probability distribution with P(x)=0 for every x. They are not impossible, only 0 likely.

But there is no constant probability distribution for only the rational numbers from this interval. Or from any other, for that matter. Nor there is an uniform probability distribution for all the naturals. Or for any infinite subset of naturals.