Of course, this assumes that "probability 0" entails "impossible". I don't think it does. The probability of picking a rational number may be 0, but it doesn't seem impossible.

Given uncountable sample space, P(A)=0 does not necessarily imply that A is impossible. A is impossible iff the intersection of A and sample space is empty.

Intuitively speaking, one could say that P(A)=0 means that A resembles "a miracle" in a sense that if we perform n independent experiments, we still cannot increase the probability that A will happen at least once even if we increase n. Whereas if P(B)>0, then by increasing number of independent experiments n we can make probability of B happening at least once approach 1.