This trouble takes many forms. The Banach Tarski paradox is just one. AC also (obviously) implies that there are sets that don’t have a volume (or area, or length).
The supposed existence of nonmeasurable sets seriously complicates analysis. (Analysis is, roughly speaking, generalized calculus.) Analysis textbooks are full of results which state that such-and-such a procedure always generates a measurable set. If students ask to see an example of one of these mysterious objects that don’t have a volume (or area, or length), the instructor is in trouble. AC tells you that such sets exist, but says nothing about any particular one of them. It’s non constructive.
In fact it can be shown that almost any set that is in any sense definable (say, by logical formulas) is measurable. For example, all Borel sets are measurable. If authors simply assumed that all sets are measurable, the average text would shrink to a fraction of its size. And they wouldn’t get into trouble – it is not possible, without AC, to prove the existence of a non measurable set.