I believe that the answer to this question is "No". But it's a question for a physicist, really, not one for a philosopher nor even for a mathematician. One can certainly describe metrics on spaces that behave in the kind of way you suggest. But whether the universe is such a space is an empirical question.
Yes, a point has length, depth, and height zero. So do two points, three points, and even as many points as there are natural numbers. But if you have as many points as there are real numbers (of which there are more than there are natural numbers), then that set of points may have some positive length, depth, or height, though it may not. (In that case, they will not have zero length, depth, and height but may have no assignable length, depth, or height.) The branch of mathematics in which such things are studied is called "measure theory".
Exactly what a point is is another question. In mathematics, points may be regarded in a wide variety of ways, as is convenient. Are there any points in space itself? That's a disputed question, and an empirical one, not one on which philosophers can pronounce.