I've thought about this same thing. To me it makes more sense to store mechanical energy as mechanical energy.
Here's the hard part in my mind. 2 things I think you would want to figure out.
First you would want some kind of transmission that would gear up your compressor to pump faster at low pressure and slower but with more torque as the pressure got higher.
Second, if you could figure out a way to fill your tanks one at a time. For example, bring the first tank up to 100 psi then the extra air would start dumping into the second tank until that one came up to pressure....ect. That way you wouldn't have to wait a week to get up to operating pressure.