We cannot store AC in batteries because AC changes their polarity up to 50 (When frequency = 50 Hz) or 60 (When frequency = 60 Hz) times in a second. Therefore the battery terminals keep changing i.e. Positive (+ve) becomes Negative (-Ve) and vice versa, but the battery cannot change their terminals with the same speed so that’s why we can’t store AC in Batteries.
In addition, when we connect a battery with AC Supply, then it will charge during the positive half cycle and discharge during the negative half cycle because the Positive (+ve) half cycle cancels the Negative (-Ve) half-cycle, so the average voltage or current in a complete cycle is Zero. So there is no chance to store AC in the Batteries.
Good to know: Average Voltage x Average Current ≠ Average Power.