How do cloves break down plastic? When my family buys powdered cloves in bulk, we put them in plastic until we get home - which takes at very most an hour. As soon as we get home, we transfer them into glass because we have had issues with them eating through the plastic cups until they are soft and have holes in them - this takes maybe a week. 

I do not know exactly what type of plastic the cups are made out of; if this helps, they have the recyclable number 6 on them.

So, my question is, what is it in the cloves that is causing them to eat through the plastic at such a high rate? Are they really acidic? Or is it perhaps due to their oils?
Most of the odor of cloves comes from volatile molecules, especially eugenol, found in the cloves.  Eugenol is a water-immiscible, oily liquid at ambient temperatures.

Plastic with resin identifiction code #6 is polystyrene. 

Polystyrene is not compatible with most organic solvents, because it tends to be softened, swelled, or even completely dissolved by them.

Thus I suspect that the problem is that oil in the cloves is attacking the plastic that you use.  You could avoid the problem (mostly) by using a plastic with resin code #2 or #5 (i.e. high-density polyethylene or polypropylene) instead of #6.  But, you are probably right that glass would be better than any plastic for storing your cloves.