My motivation for this comes from Minkowski space and general relativity more broadly, but I don't want to focus on the real world details currently because I want to understand what a positive vs negative distance between points means structurally for a space, rather than hand-waving that they indicate space-like vs. time-like separation which say little for the intrinsic meaning of metric tensors.
I want to gain an intrinsic understanding for what a metric tensor "means", and clearly the intuitive idea of "length" doesn't cut it here, given metric tensors can sometimes be negative, sometimes be positive, and also be 0 for non-identical vectors. What can be said about the "meaning" of a metric tensor which is not positive definite? Are there any familiar day-to-day objects which can be modeled by manifolds with metric tensors which may be positive and negative?