Not sure, but if you replace AC with a weaker axiom, Axiom of Dependent Choice, you can construct a model in which all subsets of reals are measurable (called

Not sure, but if you replace AC with a weaker axiom, Axiom of Dependent Choice, you can construct a model in which all subsets of reals are measurable (called the Solovay model), which precludes the Banach-Tarski paradox (which requires non-measurable sets). Could BT be constructed without AC but with other way of ensuring there exist non-measurable sets? No idea.