Yes, your power will primarily come from your batteries. When your batteries are full, extra power generated is either sold back to the grid if you live in a nice place, or credited on your bill with some limited roll over system. When your batteries are empty, you buy power from the grid. If your batteries are empty when the grid is down, then you would indeed also experience the outage.

But you can get alerts when the grid is down and choose whether you want to conserve power to make sure you don't drain your batteries at that time. You can still gain panel power during a grid outage, you just won't have a back-up source of power or the ability to sell excess.