Of course, a measure is in general only defined on some $\sigma$-algebra of sets, and for Haar measure that's the Borel sets. As Pete mentioned you can complete the measure if you like, but that still won't yield all sets. So for an arbitrary set $E$, $\mu(E)$ may not even be defined.
You can see this is necessary with an example you already know: let $G = S^1$ thought of as $[0,1]$ mod endpoints, and let $E$ be the usual example of a non-measurable set (the Vitali set). There's no value for $\mu(E)$ that will let $\mu$ be translation invariant, finite, and nontrivial, which are required for a Haar measure, and so there's no "Haar measure" defined for all subsets. To be able to have a Haar measure, we need to define it only on "reasonable", i.e. Borel, sets.