How long can I store 1 Gal jugs of distilled water?

The answer is indefinitely. As long as the seal is good so no air gets in. As per the other persons answer, leaching could be an issue depending on the type of plastic. Look at the bottom of the jug, there should be a plastic type or number in a small triangle. #1,#2, #4, #5 and #6 are ok. #3 is not because it uses a polyvinyl and could leach BPA.