This isn't even close to true. A lithium-ion battery can store roughly 1 megajoule per liter, which I'm going to use because it makes the math easier. (Other batteries are slightly less dense, but the correction factor is small). Total global energy consumption is around 1 exajoule per day. That's a trillion liters to store a day's worth of energy, or one billion cubic meters. Assuming you can stack batteries ten meters high -- which seems doable -- a battery farm to store a day's worth of energy for the whole world would take up about 100 million square meters, or 100 square kilometers, slightly smaller than San Francisco. That's less than the amount of space currently taken up by solar or wind farms.
Using zinc-bromine or other technologies will take up more space, but as you can see, a small reduction in energy density won't be that big of a deal.