Wikipedia states that the Haar measure on a compact group is a mean (and that every compact group is amenable). But, obviously, the Haar mesure on the group of unit quaternions cannot be defined on every subset, else the BanachTarski paradox would not happen. What am I missing?
The definition of a mean for a locally compact group is not a finitely additive measure defined on the entire power set of $G$that definition is only correct for discrete $G$. See the wikipedia article for the correct definition. Indeed, the definition given there refers to the Haar measure, so is easily seen to hold in the compact case. For indiscrete $G$ there will typically be nonmeasurable subsets; of course, the pieces that appear in the BanachTarski paradox are nonmeasurable. 

