If you say map instead of embedding and measure area with multiplicity, then the problem becomes more interesting.
The following analog of Nash--Kuiper theorem was proved by Gromov in his "Partial differential relations":
Let $f\colon M\to N$ be a short map between Riemannian manifolds of the same dimenssion. Then there is arbitrarily close length-preserving map $f_\varepsilon\colon M\to N$.
The provided map is evidently a maximizer. It is not smooth and typically has creases at everywhere dense subset, but one can smooth it keeping the area nearly the same.