If you are into this thing and looking for an even more stupid idea to store energy, I present to you the StEnSEA [3]. Rolls right off the tongue, right? It is a hollow concrete sphere that is lowered to the bottom of the lake. Pumps then remove water from it, creating a vacuum. Letting the water back in and using the pumps as generators, the energy is reclaimed. Curiously absent from all documentation of this project is the amount of energy stored. I did some back of the envelope calculations a while back and it is 3.8kWh, for a multi-million-euro prototype!
[1] https://www.energyvault.com/ev1
[2] https://www.youtube.com/watch?v=iGGOjD_OtAM
[3] https://www.iee.fraunhofer.de/de/projekte/suche/2013/stensea...
That seems very low. Their website mentions 20 Mwh+.
Though my back of the envelope agrees with yours.
And building concrete spheres that can withstand the pressure of a 700m water column is probably an interesting design challenge on its own.
Most proposals are even worse, as they suggest concrete rather than sand. The monetary cost works out even then, but concrete production currently emits CO2, and the combination of that with the low energy density means they'd have to run for around a century to only be as bad as fossil fuels.
https://gravity-storage.com/ are trying to make compact hydroelectric plants by putting weight on the fluid. Wish someone would give them loads of money :/