The reason we can't list the real numbers between 0 and 1 in a sequence is that there are too many irrational numbers. Nice going, guys. Way to ruin it for the team.
Think about it: we can list the rational numbers. If we could also list the irrational numbers, then we could list the real numbers by interweaving the two lists, taking one rational number, one irrational number, one rational number, etc. We might even be able to macramé ourselves an attractive lampshade.
Anyway, our lists would look like this:
rational numbers: r1, r2, r3,...
irrational numbers: s1, s2, s3,...
real numbers: r1, s1, r2, s2, ....
Since we can list the rational numbers but not the real numbers, we must not be able to list the irrational numbers either. What a buzzkill.