There is an amazing theorem of Morrey and Grauert that says that not only does every (paracompact) smooth manifold have a real analytic structure, the real analytic structure is *unique*. Using Whitney's ideas, you can show that two real analytic manifolds $M$ and $M'$ that are diffeomorphic are also real-analytic equivalent, if they both embed analytically in Euclidean space. And of course Whitney lets you assume that one of them does. The question boils down to whether real analytic real-valued functions on a real analytic manifold separate points. Morrey and Grauert proved that they do. Although I don't understand either proof, I remember that a key step in Grauert's proof is to make the tangent bundle of $M$ into a complex manifold, and show that it is a Stein manifold.

Anyway, the result is much harder than what Whitney did, which is a very good but expected application of the Weierstrass approximation theorem. To understand the issues, you can consider instead real algebraic manifolds in the geometric topology sense (rather than in the algebraic geometry sense). These are called Nash manifolds. A circle has an embeddable Nash structure, the one where trigonometric functions separate points. But $\mathbb{R}/\mathbb{Z}$ is another Nash circle, one of zillions that do not embed as Nash manifolds into Euclidean space.