Answer:
After you have cooked the barley, you must cool any leftovers to room temperature, usually while you eat. Then place the leftovers into the refrigerator UNCOVERED for 10-15 minutes, preferably spead flat on a plate, in order to cool it off quickly. After it has cooled to fridge temperature, you may pack it into tupperware. Be sure to place some plastic wrap to the top of the barley to minimize air contact, which will dry out the topmost barley. Place it back in the fridge.
After following these instructions for storage, the barley will last 3-5 days, depending on the internal temperature of the refrigerator.