Is there a notion of "space" satisfying the following requirements?

Spaces form (at least) a category; morphisms between spaces are called "continuous maps."

Every topological space is a space, and the inclusion of $\mathbf{Top}$ in the category of spaces is fully faithful and injective on objects.

The category $\mathbb{R}\mathbf{Mod}$ is a space.

If $X$ is a topological space, a vector bundle over $X$ can be described as a continuous map $X \rightarrow \mathbb{R}\mathbf{Mod}.$

(I don't mind if some of these requirements are partially violated, they're just meant to be guidelines.)