Full Idea
One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original.
Gist of Idea
The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres
Source
Penelope Maddy (Defending the Axioms [2011], 1.3)
Book Reference
Maddy,Penelope: 'Defending the Axioms' [OUP 2013], p.35
A Reaction
(The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure.