When you say 'non-integer number of dimensions', it actually sounds like you are asking about spacetime with a fractal dimension.
I know very little about this subject, but it's a fun question to think about. It goes without saying that I welcome someone with more specialist knowledge to elaborate, but here are some highly speculative remarks for now.
To my knowledge, non-integer dimensions arise from a sort of universal roughness at arbitrarily small scales. Keep in mind that on arbitrary rough surfaces, it is possible that the curvature varies so wildly that the distance integral (defined at each scale by simultaneously approximating the geodesic path at that scale and computing the associated integral bounds) fails to converge. In this case, you need a more general concept of distance, likely tailored to the geometry at hand.
The simplest approach is to define distance functions on a family of smooth (or almost always smooth) manifolds that approaches the manifold you have in mind. This family of smooth manifolds can also be used to define a rough manifold, though the criteria for whether two families of manifolds define the same limit is probably more subtle than it is for Cauchy sequences. Work along these lines has been done for Brownian paths, for example, which exhibit a fractal structure (c.f. metric properties of mean wiggly continua.) The geodesic distance between any two points could diverge in the rough limit, but there may be some structure to its divergence that you could use to define a 'fractal distance'. In the simplest example, rescaling the distance by a power of the 'amount of roughness' could lead to a well-defined distance between points. This corresponds to measuring distance in units of a fixed 'rough meter stick'.
Exhibiting a 'rough meter stick' can be difficult, however. If you can define a non-trivial Brownian motion on the rough manifold, then such a metric could potentially be furnished by considering the distribution of times taken to travel from one extended region to another (likely defined in a limiting sense), and taking the limit as the regions shrink to two points. The shape of the distribution function may change significantly depending on the location of the points, but could have universal characteristics as the 'distance' between the points is made arbitrarily small.