Scientists propose converting tall buildings into batteries

Simplifying, any of these are a winch, any number of pulleys and support for them.

Elevators normally are "direct drive", i.e. the winch is right over the shaft.

In cranes there are normally a few pulleys to "move" the cable from the winch to the lifting point, but - additionally - cranes use sets of pulleys to demultiplicate the tension on the cable.

Example: https://www.zoke-crane.com/posts/2708/

the more turns around pulleys you have the more you increase the lift capacity BUT at the same time you slow down the speed.

In normal operation of cranes you change the way the cable goes around the sets of pulleys, if you have less weight to lift you use a more direct drive (and have more speed) if you have more weight to lift you use a less direct drive (and have less speed).

In theory if an elevator winch (and cable) can lift 1 ton at (say) 3 m/s, it can also lift 2 tons (but at 1.5 m/s speed) by using a set of pulleys (it will also need double the length of cable, that the drum may not be able to receive) BUT you cannot change the cable path on-the-fly, and surely the beams that support the winch were designed for 1 ton, not 2.

So you could devise a smart way to have small (1 ton each) containers of some heavy material (of course not wet sand, which is in practice the worst cause of corrosion for metal) more likely cubes made of concrete be loaded and unloaded (and translated horizzontally to and from storage).

It is not like you can put that many of these 1 ton cubes on the highest floor of a tall building without structural consequences for the building, so it is not just a matter of replacing/retrofitting the elevator winch.

There is a very clever similar idea/experiment using a crane and concrete blocks:

https://www.wired.com/story/energy-vault-gravity-storage/

but at least they built a dedicated plant.