I think it's more likely we'll see an asteroid mining economy bringing lithium for batteries on Earth than millions of 1km holes with 20 ton weights storing energy for single charges of electric vehicles.
Whatever local (i.e. small scale) energy storage we have in the future you can probably bet (from first principles) with 99% certainty it will be based on the electromagnetic force. Gravity is too weak to be practical for smaller scale storage and cannot give mobile storage units since you have to deal with huge weights. For large scale see the "pumped storage" system already mentioned in this thread. Just for illustration, the "Taum Sauk Hydroelectric Power Station" can produce 175MW of continuous power. If you dropped the 18T weight in the 1km shaft you'd get about 13MW averaged over the 14s fall. Then you have to lift it up again if there are still enough pieces left.
Even if you found a way to store energy in nuclear interactions (i.e. "charge a nuclear battery") you don't want to have a bunch of containers full of radioactive material all over the place. If one thought it through, I wouldn't be surprised if there are in-principle issues for the charging part similar to the ones with gravity.
Since we don't know of any others, this leaves only the electromagnetic force to store energy with sufficient mass (or volume) density in practical ways. Whatever it ends up being (chemical batteries, supercapacitors or something else) the future local energy storage unit will separate and hold charges apart.