By de Morgan's law it is the intersection of the complements of spheres with irrational radius.
So it is exactly the set of points whose distance from the origin is not an irrational number. But then the distance must be a rational number $q$ and such a point is then in $S_q$.
If $U$ is an open set that includes $E^C$, then $U$ can certainly have infinite measure. For example, $U=\mathbb{R}^2$ would work. The point is that even though $E^C$ is a dense subset of $\mathbb{R}^2$, it need not have large measure.
A common confusion of beginners is that open dense sets must be the whole space or almost the whole space.
@BAYMAX Under Lebesgue measure, every nonempty open set has positive measure. But open dense sets can have arbitrarily small measure, as in the example I gave. A great thin book that relates measure and concepts such as being open dense is "Measure and Category" by John Oxtoby.
