I'm going to disagree with the other answers: I think that the angular momentum of macroscopic "classical" objects is not quantized.

Consider an automobile tire spinning in a wheel well. On the tire is a device that triggers whenever a certain point on the wheel crosses a certain point on the well, adding one to an internal counter if it passes it clockwise and subtracting one if it crosses it counterclockwise. (Alternately, you could dispense with the wheel well and say that the device tracks its own position by inertial navigation.) The state of this system can be described by a value $θ\in\mathbb R$, where the current value of the counter is $\lfloor θ/2π\rfloor$ and the angle of the wheel is $θ\text{ (mod }2π\text{)}$. In the absence of external forces, the Hamiltonian of the system is essentially that of a free particle in $\mathbb R$, and the spectrum of angular momenta is continuous just like the free particle's momentum spectrum.

That's a 2+1 dimensional system. In 3+1 dimensions, there's the Dirac belt trick to worry about. Does it matter? I don't think so. There's no reason to limit the device to holding a single integer, or to being reversible. It could simply store the entire history of its orientation readings internally, or broadcast them by radio, indelibly recording them in the universal wave function. That's a very noncompact state space, and it's an accurate enough model of bodies like the earth.

The angular momentum operator on this monstrosity obviously violates the assumptions of any proof of the quantization of angular momentum, but that's no reason not to call it angular momentum. We do call it angular momentum, and it's what the question was about.