Can I switch out the hard drive from one Windows 7 machine

[quixotic1]

7. Careful...it's actually harder than you think

Seems like that would depend, in part, on the fan that's failing. If it's the power supply fan, then yeah...that's a bit of a chore, even if you're using a standard case. (Also, could depend on the manufacturer.)

However, the reason it might not be quite so easy to swap a hard drive is because Microsoft collects information about the system running a specific Windows license. (This comes, in part, from attempts to reduce the sharing of activation codes/keys.) A fingerprint is calculated and, if it varies too much from the original activation fingerprint, the license validity may be questioned.

Even if the machine is the same model, there's no guarantee that the internal hardware is the same. (I've seen components vary between machines delivered in the same batch.)

Depending on the original manufacturer, the fan might actually turn out to be far less work than you think. In general, it should be attached using a couple of screws (pretty easy) and connected to the motherboard using a standard clip that slides onto the pins. (There might be a power supply connected as well.) The "trick" is making sure that you connect the new fan to the same pins used for the old fan.