It means you either need an alternative when production is too low such as coal or gas-fired power plants or a lot of capacity sufficiently stretched out than they are not stopped at the same time. Managing such a large grid with huge swings in capacity and making it resilient is a massive challenge. That’s why you end up with Germany building 70-ish new gas-fired power plants next to their alleged push towards renewable.
It’s probably doable but when you look at it from this angle nuclear starts to look good as an alternative.
Batteries aren't the only storage. The better options in my opinion are the places where you can use the landscape to your advantage. Pump a lake full when there's too much power and let it drain when there's too little.
Also in a connected grid setup, the sun always shines somewhere though that does come with potentially huge transmission losses from distance
We already do that. France notably has a lot of hydropower and they pump water up when they don’t want to shutdown a nuclear unit.
The issue is that there is very little places where you could build new dams in Europe and water shortage is becoming a regular occurrence.
Yes water shortage might be a problem if the river you're on runs dry. That's not often a problem though, plenty of major rivers. And a dam doesn't change the total amount that flows, it just changes when. As a result it might even help in lowering some flood risks.