  We prove that a nonempty, proper subset $S$ of the complex plane can be
approximated in a strong sense by polynomial filled Julia sets if and only if
$S$ is bounded and $\hat{\mathbb{C}} \setminus \textrm{int}(S)$ is connected.
The proof that such a set is approximable by filled Julia sets is constructive
and relies on Fekete polynomials. Illustrative examples are presented. We also
prove an estimate for the rate of approximation in terms of geometric and
potential theoretic quantities.
