A good rule of thumb is the size of the battery should be proportional to the load applied to it, and the load applied to it should be no more than the maximal charging/discharging rate.
Suppose for example you settle on flooded lead-acid as a battery chemistry. That's what I'm using for my well-pump. Assume the maximal charging/discharging rate is 1/8th of C, with C being the 20hr battery capacity. Assume it's a 24V system using the Conext 4024 inverter.
So, that means you need a battery that is at least 8 times bigger than the amp load pulled out of it to run your pump. If your pump draws 960W while running, the amp draw out of the battery would be 960Wh/24V = 40Ah. A battery that's 8 times bigger than 40Ah would be 40A X 8 = 320Ah. A Trojan L-16 battery is typically in the 380-420Ah range, so four 350$ Trojans would fit nicely here.
Suppose instead you went with Li, with a max charge/discharge rate of 1/4th C. Then the battery size would be 40Ah X 4 = 160Ah.
It's always better to have a bit more than a bit less, so I'd recommend getting a lead-acid battery in the 400Ah range, or a LiPO4Fe battery in the 200Ah range. The bit extra covers system losses/inefficiencies.
With 2X solar capacity, you are looking at the 4024 inverter mated to a 200-400Ah battery at 24V and 1800-2000W of solar panels. You'll need a charge controller that can handle 60-80amps at 25-26V charging. I have a Midnight200V model for my system. You might also look at Epever's Tracer series, maybe the 6420AN or the 8420AN. That would allow you to wire eight 250W grid-tie panels 4S2P.
