Every closed orientable (smooth) $n$-manifold (smoothly) embeds in $\mathbb{R}^{2n-1}.$ This is true for $n> 4$ by Haefliger-Hirsch, $n=3$ by C.T.C. Wall (Wall's paper has the reference to Haefliger-Hirsch)
Wall, C. T. C.
All 3-manifolds imbed in 5-space.
Bull. Amer. Math. Soc. 71 1965 564–567.
For $n=4$ this is true in the topological category, and is open (as far as I know) in the smooth category [which means that no counterexample is known] - see Bruno Martelli's answer to this question.
