The Nash-Kuiper embedding theorem states that any orientable 2-manifold is isometrically ${\cal C}^1$-embeddable in $\mathbb{R}^3$.
A theorem of Thompkins [cited below] implies that as soon as one moves to ${\cal C}^2$, even
compact flat $n$-manifolds cannot be isometrically ${\cal C}^2$-immersed in $\mathbb{R}^{2n-1}$.
So the answer to your question for smooth embeddings is: *No*, as others have pointed out.
I believe Gromov reduced the dimension you quote of the space needed for any compact surface to 5,
but I don't have a precise reference for that.

Tompkins, C. "Isometric embedding of flat manifolds in Euclidean space," *Duke Math.J.* <b>5</b>(1): 1939, 58-61.