Consider a metric space with a path between any two points, so a real line segment of some length between them, and the length of this line is the same as the distance between the two points in the metric. This is called a geodesic metric space. Any two points are connected with a shortest path. There could also be multiple such paths as with the antipodes of a sphere. What I mean by homogenous is that any point can be mapped to another while preserving the structure of the space. Another way of looking at it is that the space is the same no matter what point is taken as the origin.
Geometries like Euclidean and non-Euclidean geometry would be examples. However, I am curious about examples which are not locally Euclidean. So, they would behave kind of like Euclidean space in that there is a measurable shortest path between two points and everywhere in space is the same. So you could "move around" in a given "direction" in this space, but it does not locally resemble Euclidean space. Is this even possible? If so what characteristics would these spaces have and could they be classified? What about the underlying topologies of these spaces?