Gigawatts are a unit of power, not energy. Storage is measured in gigawatt-hours. Stanford's PR department should know this. The question is, how much storage is needed to make it through the night?
The biggest pumped storage station in the US [1] generates 2.7GW of power and stores 24GWH. So, two of those, somewhere in the Sierras, should cover a night of charging.
[1] https://en.wikipedia.org/wiki/Bath_County_Pumped_Storage_Sta...