Can we make a rechargeable zinc air battery at home?
According to this video it is possible to make a zinc-air battery. I just don't know if it is rechargeable.

You just need a zinc sheet for the negative electrode, and a steel wool for the positive electrode separated by an isolator soaked in sodium hydroxide as an electrolyte.

Update

I've been reading a little about zinc-air batteries and I can confirm that this configuration does not produce a rechargeable battery.

The rechargeable zin-air battery technology is a very recent one and I don't believe that we can find anything about how to do it at home in the recent future.