I don't think I've seen a definition of a space-like notion phrased only in terms of paths, but you could certainly write one down, perhaps as an example of concrete sheaves. It seems related to the notion of Froelicher space which defines "smoothness" in terms of paths and "co-paths."
However, it seems unlikely to me that you'll get a very good notion without including also some higher-dimensional test objects in addition to intervals. For instance, there is the notion of $\Delta$-generated space, which can be described as a set $X$ with sets of distinguished maps $\Delta^n \to X$ for all $n$, satisfying a bunch of properties. I don't know any description of those properties other than "if $X$ is given the final topology induced by the distingished maps, then all continuous maps $\Delta^n \to X$ are distinguished", but that doesn't mean there isn't one; I don't know whether anyone has looked.