Banach Tarski, in some sense, really can't make sense in the universe because the Universe is... discreteish, as you can't get smaller than the Planck Lenth (1.6 x 10^(-35)), and while that's very small, as far as maths is concerned that means the number of "identifiable points" in a sphere of any finite size is itself finite, not even a countable infinity.
I realise that's all very dodgy, but of course we are also bringing in quantum mechanics -- but I consider it enough to say Banach Tarski doesn't apply :)
My understanding is that Planck Length is not a limit to the smallest possible thing; it's a limit to the most precise possible measurement. So there could in fact be lots of things happening at sub-Planck Length scales, but we could never observe them.
I realise that's all very dodgy, but of course we are also bringing in quantum mechanics -- but I consider it enough to say Banach Tarski doesn't apply :)