> You need to have batteries that can store power for at least a week to have base load as reliable as nuclear power.
This is a complete myth, somebody pulled "a week" out of their butt a decade ago, it gets repeated a ton, but it's not based on reality or studies or numbers. This is a consistent problem with online nuclear advocacy: there's no basis for the numbers, nobody calculates anything, and if they bother to do a calculation they only calculate the upper bound and then assert "see look a big number" and say that's a proof of impossibility.
What event requires a week of storage? Nobody can name one! When has there been a week with zero generation? No one can name it! The assumptions that one has to make up in order to make a "week" even sound plausible are in turn themselves so implausible.
> There isn't enough battery capacity in the world to do this for a state like California, let alone the whole country.
Imagining there's a fixed battery capacity is a very short sighted view, it's growing by 10x every year.
So let's take your "week" as the measure, even though it's wrong. If we're at 2-3 TWh of world battery production capacity in 2025, that's 4 days of California demand. By 2031 or 2032, we're going to have 20-30TWh of battery production.