If you are trying to use your domestic clothes dryer, look at the plate in the back; it will tell you how many kilowatts the power source has to be capable of. It is about watts, not volts. A battery bank would be that many watts capable for the hours of operation, plus conversion losses.
There is little 12V and large 12V batteries.
If your dryer runs on 240V, twenty - 12V batteries can yield 240V, but the motor/timers would have to be modified to run on battery DC, not AC. That battery bank has to be recharged anyway, with the use of more electrical power than the AC would supply for the dryer.
And the batteries + inverter will cost you more than the power for several years running the 'normal' dryer.
You cannot win.
To dry clothes indoors, takes power and time. You can reduce one by increasing the other, but the multiplication will yield the same wattshour.
If you want to MAKE a clothes dryer that will take less power, a large enough 'box' /'closet' with all clothes hanging could work if you use only a fan into it and exhaust the moist air, perhaps running overnight... or longer. Would only use the power needed to run the fan for a certain time and no heat added.
It would be like 'importing' a piece of not-sunny breezy outdoors into the basement.
90% of the planet population does it 'green' outdoors. What is the problem with that ? If you are after the 'comfort' an indoor dyer provides, that will cost you; so give up the comfort if in need to save money. For an apartment I would do the 'air box'