Yea, for the most part, my apt is pretty quiet unless the HVAC is running. Would some rubber washers work to dampen the sound, or is there some other system I should use?
First thing to do is really sit the HDD on a block of foam inside the PC. This will isolate the drive from the case and eliminate the vibration noise. If this test is successful in noise reduction then your HDD is making the case vibrate and resonate causing noise.
The most common way of reducing the HDD vibration is rubberised mount systems. They can be quite effective depending on the brand (different qualities) and how tight you fix it to the PC. The tighter you screw the drive in with these, the more compressed the rubber is and the more vibration travels into the case and makes awful noise.
Leaving the HDD on the foam is another option. It should be safe there if you are not moving your PC around or it wont be knocked. Temperatures in this setup will more than likely be higher but in most cases should not be alarmedly increased.
My method and the one favoured over on the Silent PC Review (SPCR) forums is a form of elastic suspension. Take normal trouser elastic purchased from a local store, thread it around the case and tie in the Hard Drive. When done correctly, the HDD's are very isolated and pass near zero vibration onto the case. In my PC, the drives would be ok for hand transportation but I would not trust the elastic to cradle my drives during a car journey.
You can also purchase noise isolator caddies for hard drives. Essentially encasing them in a manufactured enclosure which reduces the vibration and noise reaching the outside world. These can suffer from heat problems and it would serve you well to consult SPCR before going to purchase one.
As is typical in most troubleshooting, you need to isolate your issue and choose the most appropriate method for you to resolve it. Hopefully I have provided I have provided you with a good basis to start from.