No, batteries are not needed. Your appliances don't turn on and off. The inverter is in phase with the line coming into your house.
I think an easy analogy is with water. Imagine you're hooked up to the city's water supply. And, you drill a well and put in a water pump in your basement. Let's say that at time A, you're using 10 gallons per minute (LOL) in the shower, but your well pump is only providing 9 gallons per minute. The city water will pump 1 gallon per minute in, which you will be billed for. Let's say at time B, you're using 5 gallons per minute, and your well is pumping 7 gallons per minute. Your extra 2 gallons per minute flow out of your house into the city water system, and your meter "runs backwards." But, the water from the city and water from your well is identical. Your shower can't tell the difference, nor can it tell the difference which source it came from. (There are two different types of metering in this case, some that do, more or less, run backward, and some systems have separate meters for the amount that goes out, and the amount that comes in.)
Also, batteries can add a huge expense to the system. While the photovoltaic panels have lifespans of 25 or more years, the batteries have lifespans far less than that. E.g. a decent marine battery runs around $100. At a guess, you'd need a couple dozen of them, and would replace them roughly every 5 years (wild guesses by me).