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 Banach-Tarski 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 non-measurable subsets; of course, the pieces that appear in the Banach--Tarski paradox are non-measurable.