There seem to be a lot of approximations with filters. How sharp can the actual ``passing'' zones be made?

In principle, you can make the passing zones arbitrarily sharp by adding enough reactive components with appropriate impedances to make steep slopes. If you want a sharp cutoff so that zero signal gets through for a range of $\omega$'s, you need a lot of zeros (in fact, an infinite number in the limit of perfect suppression) covering a whole region along the $\omega$ axis in the ${\bf s}$ plane. Since you need at least one reactive element per zero, this could be difficult in practice.