Not all things are possible, only potentially possible ones.
If you take all the numbers between zero and one, you will find an endless supply of distinct numbers but none of them are 3.
Aren't you really just talking about the level of abstraction being applied? If there are infinite possibilities between zero and one, then it's just our way of relating to it that imposes the perception of a limitation.
Given infinite time, is it possible to randomly draw a rational (instead of an irrational) from the interval [0, 1]? And if so, is it possible infinite times?
In this case it does. Describe any effective measure to pick a random sample from the interval and you will have cheated into choosing from the computable reals instead (of which there is a countable amount)
Disclaimer: total armchair philosophy.
-- David Hilbert.