You're asking for quite a lot of battery. Let's say you want to use car batteries, since they are cheap and easily available. From horta's answer, you want a total of 1600 watt-hours. 1600 watt-hours / 12 volts = 134 amp-hours. Furthermore, you do not want to discharge a car battery to less than 50% of capacity, so your actual capacity should be 268 amp-hours.
So, how much capacity does a car battery have? Well, the appropriate number is RC, reserve capacity minutes at 25 amps. If a battery has an RC of 60 minutes, it is a 25 amp-hr unit. And this is about what you get at the low end of battery sizes, so I'll use it as an example. 268 / 25 = 10.72, so for this size battery you'll need 11 batteries. As you can see, you'll need quite an installation. And, as I'll discuss later, you need at least a day's extra capacity, so you'll actually need about 22 batteries. If you use bigger batteries (and I've seen ratings as high as about 150, you'll need fewer, but they'll cost more per battery.
And then there's the solar cells you expect to use. In the long run, battery charging is about 60% efficient, so you'll need a total daily solar panel output of 1600 / 0.6, or 2700 watt-hours. This will take place over a period of 8 hours, but most of the time you're getting way less than peak power, so let's size the array for 4 hours. 2700 watt-hours / 4 hours is 675 watts. And that assumes that every day is sunny. At a minimum, you need to assume at least one day's worth of reserve, so your minimum solar size is 1300 watts. Just as an example, Harbor Freight is selling a 45 watt panel kit, with a size of 3 ft x 3 ft. So just as a rough number, you'll need 2700 / 45, or 60 panels, with a total area of about 540 square feet.