As a answer, first I'd like to ask why you're asking a quantum mechanics problem with classical mechanics mental model.
There are two types of angular momentum in quantum mechanics:
Orbital angular momentum, which is a generalization of angular momentum in classical mechanics (L=r×p). I think, you shouldn't have problem with this because Orbital has size.
Spin, which has no analogue in classical mechanics. You can understand it as a number appeared in quantum equation. It can be understood like charge, which is a number (with physical dimension) to, which is a number to denote one of basic attributes of particles. Yes, Spin does have physical dimension of angular momentum. But, its because it is a type of angular momentum, mathematically.