Of course, in most states there is net metering, so it is foolish to store it; feed it to the grid. Storing in a battery will cost you about 15 cents/kWh and more like 30 cents/kWh with Tesla powerwall, so you need to have that large a difference between the cost of power when you store it and when you take it out. It is unusual to have a battery to store all your unused solar output, that would be very expensive. You don't want to store more than you will actually need, and in particular more than you need from 3pm to 9pm or whenever your local high-peak rate is. So it's up to you to measure how much you need then. (You will have solar from 3pm to 5pm or so, a bit later in high summer.)

If you seek home backup, generally you don't want to pay to run your home at full load during an outage. Surely in an outage you won't charge your car, or run your over or dryer, or your air conditioner on full etc. Consider how much power you need when conserving, unless you are expecting an apocalypse style outage. Get enough battery for that.