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Or you could avoid building the machinery and just pump water up and down. Then to lower the cost you could drop the tower idea and use a natural hill with reservoir on top. Surprise, surprise you just invented a pumped-storage power station.

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Pumped hydro is great, but there are relatively few geographic areas where the terrain is suitable. You need a valley that can be dammed to fill a reservoir, enough water that it actually becomes a reservoir, and then a gentle enough slope coming out of the dam that you can build a secondary reservoir with enough water to capture the outflow of whatever period of time you're storing the energy and pump it back in.

California is one potential location, and there's been a lot of interest in pumped hydro from the California Department of Energy (and utilities), but I doubt it'd work as well nationwide.

Dig a 1300m shaft in the ground. Robots would be ideal for construction, as a large cistern would need to be built at the bottom.

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