There are a number of processes happening which are causing your rubber bands to deteriorate. One perhaps surprising cause is that rubber bands are affected by ozone in the air, to the extent that rubber bands are used to test the amount of ozone in the air. Ozone and rubber deterioration
Ozone is naturally found in air, particularly as a result of a lightning strike or other electrical activity (e.g., high voltage equipment, fluorescent lighting). Ozone is responsible for the "fresh" air after a storm. The only way to isolate your rubber bands from ozone is by storing them in an airtight container.
If you have a large bag of rubber bands, you should break them into smaller collections in airtight containers, and use up all the rubber bands in one of the smaller containers before opening another collection. That way, the rubber bands are exposed to new air for a shorter amount of time before they are used. That will also reduce the effect of ordinary oxidation on the rubber bands (which is probably less significant than the effect of ozone, depending on the exact composition of the rubber bands).
Another effect is the loss of volatile organic compounds (VOCs) from the rubber bands. Keeping them in a small container might help with that (the VOCs exchange between the rubber bands and the air in the container might reach equilibrium sooner in a small container?) but probably a better solution is to keep the containers you are not using somewhere cool (basement, refrigerator, or freezer).
Finally, UV light will have a negative effect on rubber, so you should keep your rubber bands in the dark.
You can pay more to get rubber bands with a UV and Ozone resistant coating, which may be beneficial depending on the application.