In general, I would expect this to be a quite intractable problem. For instance, let's assume we are only interested in the category of manifolds, and we ask the question which $3$-manifolds are covered by $\mathbb{R}^{3}.$ Here, by the solution to the geometrization conjecture for $3$-manifolds, every closed, orientable, irreducible and atoroidal $3$-manifold with infinite fundamental group is hyperbolic, and is therefore covered by $\mathbb{R}^{3}.$ In fact, most of the $8$ model geometries in $3$-dimensions are diffeomorphic to $\mathbb{R}^{3}$, with the exceptions being $S^{2}\times \mathbb{R}$ and $S^{3},$ and therefore in a sense which I won't make precise, the building blocks of "most" $3$-manifolds are covered by $\mathbb{R}^{3}.$ The sense in which you call this a classification of $3$-manifolds covered by $\mathbb{R}^{3}$ is up for debate.

In general, every closed aspherical manifold has contractible universal cover, of which $\mathbb{R}^{n}$ is the usual candidate, so hoping to classify the topological spaces covered by $\mathbb{R}^{n}$ would include a classification of "most" aspherical manifolds.

You might ask for a much coarser classification, but this at least shows that the problem is very complicated.

Spaces of Constant Curvature, part III, and indiana.edu/~jfdavis/books/Spherical_space_forms.pdf $\endgroup$ – Gro-Tsen Nov 19 '17 at 10:33