Banach Tarski, Axiom of Choice and non-measurable sets

Question

I had a question about the link between Banach Tarski paradox, Axiom of Choice and non-measurable sets. I know that the construction of sphere decomposition in the Banach Tarski context is made by the following : we take an arbitrary point and then rotate the sphere north, south, west and east to access to countably many other points that are given a coordinate such as NWSENNES and separate in 4 sets the words starting with N, the ones with W, S and E. Since, this does not reach all the points, we repeat the process for a new origin. We do it again and again and the only way to do it is to use axiom of choice so are there uncountably many origins ? I also know that every countable set (such as the words starting with N) has measure zero. Using additivity, I am tempted to say that this construction has also measure zero then. But additivity works only for countable number of sets, so I guess that the construction is made of uncountably many countable disjoint sets. I still do not understand how can a union of sets of measure zero be a non-measurable set...