This isn't an actual characterization but after Theorem 2.14 in Gilbarg and Trudinger, they also go on to remark that the boundary value problem is solvable (in 2 dimensions still, where we can use complex analysis) in any bounded domain where each connected component of the complement consits of of more than a single point.

Is this really 'dimension'?

This is indeed quick to write down and seems to be the kind of answer sought. You have written it down very neatly, too, but (and this isn't an attack on your answer) for me it is still more obvious that you can write reals in binary than the fact that "A sequence of natural numbers denotes a real number in a unique way using continued fraction." I don't see how one is "easier to define explicitly" than the other really.

Use tan/inverse tan to go between (0,1) and R. That would simplify things a fair bit.