Credit to quubog on Reddit.
distance is defined as the norm of the displacement vector. As such it cant be negative if your metric is positive definite.
Now if your metric isnt positive definite, like say in Minkowski spacetime, you can get negative norm distances, and they have a pretty clear description in that context. Say you take the metric to be (1,-1,-1,-1) then differences which have negative norm are called spacelike, means that to reach between two such points you'd need to travel faster than light, and hence causally disconnected.