If I make a target, but instead of making it a circle, I make it an immeasurable set, and you throw a dart at it, what's the probability of hitting the target?
If you construct a set in real life, then you have to have some way of judging whether the dart is "in" or "out". I reckon that any method you can think of will in fact give a measurable set.
Alternatively, there are several ways of making all sets measurable. One is to reject the Axiom of Choice. The AoC is what's used to construct immeasurable sets. It's consistent in ZF without AoC that all sets are Lebesgue measurable.
If you like the Axiom of Choice, then another alternative is to only demand that your probability measure be finitely...
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.