What kind of oscillator should I use for a binary clock?
a lot of clocks use a 32.768KHz crystal. I'm curious why that is exactly. 


This particular crystal is optimized for use in clocks.  The low frequency operation consumes less power than oscillators that run in the Megahertz.  The small size, unusual for a low-frequency crystal, allows it to be used in watches.  And the chosen frequency can be repeatedly divided by 2 until an exact 1-second time period is obtained.

The extra load capacitance isn't required to make it oscillate.  It's required to bring the crystal exactly on frequency.