The weird thing about Unity is that some incredibly complex tasks--such as adding realistic physics to objects--are just a matter of a few clicks, whereas relatively simple tasks--like making a drawer open--can end up being huge ordeals.
Case in point...well, making a drawer open. Unity does this wonderful thing where animations always try to animate from the same position. So if you have a simple back and forth drawer opening animation along the x axis, you can't simply reuse it on another drawer that's below it. Because it will try to reset the z and y axis.
The only way to solve this seems to be either making a separate animation for each object--no, I refuse--or parenting the object you want to animate with another object, and animating the parent instead. And even this solution seems to be finicky at best.
I managed to get my set of drawers working, but I'm a little disappointed that this hasn't been improved in Unity 5.
Oh well. Next step, modeling a faucet.