A positive answer is proved in S. Wagon's book "The Banach-Tarski Paradox", Theorem 13.2. Specifically, the statement proved there is:
Con(ZF) $\leftrightarrow$ Con(AF + DC + GM),
where GM is the existence of an isometry-invariant measure on all subsets of $\mathbb R^n$ taking the value $1$ on the unit cube.