Shutting down solar is super easy. You just need a switch. Wind is a bit more complicated, but it is basically stopping the rotor. The reason for negative prices are subsidies. So they can sell to the government or get some extra money as to be able to operate them properly.

Also we do not need to store insane amounts of electricity. As soon as your grid is large enough weather balances itself out fairly well. For the EU the worst production of solar and wind combined was still 897GWh in a day last year. The average was 1770GWh per day. So worst case it was half the average prodcution. If you go weekly it is 9335GWh and 12423GWh respectivly so even less. So you really only need a good enough grid and something like a days worth of storage. That actually ends up being pretty reasonable, as soon as you consider stored hydro and other flexible electricity generation.