Let $E$ be a countable dense subset of Hilbert space $l^2$. A singleton has Hausdorff dimension $0$, so $E$ has Hausdorff dimension $0$. The packing dimension of $E$ is the same as the packing dimension of the closure $\overline{E} = l^2$, namely $\infty$.
Here, $E$ is separable but not complete and not compact.