Is the angular momentum of the ceiling fan quantized? According to quantum mechanics, any quantum angular momentum is quantized in units of $\hbar$. Does it mean that the angular momentum of the ceiling fan (due to its rotation) is quantized? If yes, what does it physically mean? Does it mean that it cannot rotate with arbitrary speed?
 A: Yes, the angular momentum of the ceiling fan is quantized, in a technical sense. But in such a big system, with a macroscopic angular momentum, the possible values are so close together that it's indistinguishable from a continuum.
A: Yes, the angular momentum of a ceiling fan is quantized. This means that when the ceiling fan speeds up, it is actually jumping from one speed to another. However, the size of these jumps is so small--because Planck's constant $h$ is so small--that the difference between two allowed speeds is immeasurably small.
This is similar to how a thrown baseball's position is uncertain because of Heisenberg's Uncertainty Principle. Same as the ceiling fan, the smallness of $h$ makes the size of the uncertainty immeasurably small.
There are macroscopic systems where quantized angular momentum can be observed. When liquid helium is cooled enough to become a superfluid (~2 Kelvin), it will not rotate if the container is rotated slowly. If the containers rotation is slowly sped up, there will be a certain speed where a little whirlpool suddenly appears. The liquid helium has gained one unit of angular momentum. As the container continues to speed up, more of these quantum vortices appear, each one containing a quantized unit of angular momentum.
A: This is like asking if water is quantized.  Of course! : the basic unit of water is the $H_2O$ molecule.  Naturally when you water your plants don't think of water as "quantized": you think of it as a continuous fluid, because the basic unit is so small that it makes no practical sense to count the water molecules needed to properly dilute your fertilizer.  If anything the number of molecules is not exactly constant either (due to evaporation, humidity in the air etc).
Likewise the basic unit of angular momentum is so small and common objects have so many of these basic units that it's not practical to keep track of angular momentum this way.  Moreover, there are air molecules hitting the fan and transferring angular momentum, and other outside factors that make this number of units not exactly constant in everyday situations.  
A: Just to conjecture a contrarian opinion, no, you can't say that the angular momentum of a macroscopic fan is quantized by any quantum-mechanical stretch of the imagination.
A macroscopic fan (or any other object) is a hode-podge, extremely decoherent composite of many, many (on the order of, say, $6\times10^{23}$) microscopic (sub)systems. As such, there is no single quantummechanical observable corresponding to its angular momentum.
What you observe instead as the fan's angular momentum is just an ensemble average from many, many microscopic interactions between the fan and your measuring apparatus.
For example, suppose you rolled a single die (one of two dice) $6\times10^{23}$ times, and averaged all those outcomes. Each roll is "quantized", with possible "eigenvalues" $1,2,3,4,5,6$, so to speak. But the average will be something like $3.nnn\ldots$ with some $24$ digits after the decimal point. Hardly quantized.
And the macroscopic fan situation is even worse. Although the fan as a whole is a decoherent hodge-podge composite, there are no doubt many microscopic localized regions containing entangled superpositions of neighboring particles, just by happenstance. And then, calculating probabilities for their stochastic outcomes -- comprising an "epistemic mixture" (the dice analogy) of "ontological superpositions" -- gets even goofier.
And if that's not bad enough, the fan material likely has some crystalline/lattice/etc structure, making the situation just that much more worse, so to speak, e.g., https://en.wikipedia.org/wiki/Crystal_momentum although I have no idea what some corresponding angular momentum calculation would look like.
So one way or another, it's very, very likely that somewhere along the line, the overall system transitions from a discrete to a continuous spectrum. Thus, to repeat, the answer to your question is no (like the sign says, "Don't even think about parking here!":)
