Banach-Tarski is not absurd. Supposing the world should behave in a certain way because there is a mathematical theorem that seems to say it should, that is what is absurd.
Banach-Tarski is important and interesting, because it neatly illustrates why mathematics that seems to be sensibly related to reality sometimes does not describe reality very well. Relations between mathematics and reality are tricky.
BTW, the finiteness of particles is not sufficient to explain the problem. E.g. electrons and photons do not have a finite size. Yet you cannot something similar to a ball of photons or electrons either.
It's not just that the particles have to be zero size. There also have to be infinitely many. Otherwise, you could just count before and after and apply conservation of mass. (This also applies in mathematics: any finite set has its size preserved by euclidean motions, unless you put points on top of each other.) Needless to say, gathering an infinite number of electrons or protons will cause black hole-themed "reality breakdown" problems long before you get there.
Banach-Tarski is important and interesting, because it neatly illustrates why mathematics that seems to be sensibly related to reality sometimes does not describe reality very well. Relations between mathematics and reality are tricky.
BTW, the finiteness of particles is not sufficient to explain the problem. E.g. electrons and photons do not have a finite size. Yet you cannot something similar to a ball of photons or electrons either.