You actually don't need anything crazy like that, as you don't need continuous 200GW forever, only for 2 minutes. You just need enough energy storage so you "only" need to store 6.6 gigawatt
hours.Lithium ion battery energy density is 265 watthours per kg. So you will need 24.9 million kg of batteries. Or 24.9 thousand tons. It's a lot, Tesla factory makes more per year.