Is the closure of a set equal to the closure of the closure of that set?

Yes. $\overline{\overline A}$ is the smallest closed set containing $\overline A$. The smallest closed set containing $\overline A$ is $\overline A$ since $\overline A$ is closed.

The closure $\mbox{Cl}(A)$ of a a subset $A$ of a topological space is the intersection of all closed sets that contain $A$. If $A$ is closed, then $\mbox{Cl}(A) = A$, because $A$ is itself a closed set that contains $A$ and there can be no smaller closed set containing $A$. $\mbox{Cl}(A)$ is closed for any $A$, because it is the intersection of a family of closed sets. Hence $\mbox{Cl}(\mbox{Cl}(A)) = \mbox{Cl}(A)$.