Still, there's no physical assumption that is violated by considering the Axiom of Choice valid... whether or not the Axiom of Choice is true, you're physically not allowed to cut objects up into pieces so pathological they have no measure.
Although I know this paradox is "evidence" against the Axiom of Choice, IMHO there's a certain inconsistency of reasoning to find it troublesome that objects outside the realm of measure theory behave differently than objects inside the realm of measure theory.
To put it in perspective, how many people complain that the axiom of infinity allows us to create a nonempty set that can be divided into two disjoint subsets both the same size as the original?
Anyways, isn't the axiom of choice used in Quantum Mechanics to assert the existance of bases for uncountably infinite dimensional vector spaces? (I've had trouble digging up the precise formulation, so I'm not sure)