Here's a Q&A I found on the net:
Question: I am planning to set up a battery bank and inverter for use when the power fails. How do I calculate how much battery capacity I need? For example, if I want to run a 110 volt air conditioner that draws 7 amps for 10 hours, how do I determine how many amp hours, or cold cranking amps, I need?
Answer:
...which is why most people dont do battery backup...
Most appliances and batteries state power requirements and capabilities in amps. The problem is that you are comparing devices that supply power at differing voltage levels, 120V for US appliances, and 12V for your standard automotive battery. You can do the calculation, you just need to remember that power = voltage x current (P=VxI). If you convert everything to power (in Watts) then you can do the comparison.
Consider the air conditioner you mention in your details, 120V x 7A = 840W to power it.
For batteries the power delivery capability is stated in two specifications, amp-hours, and for automotive batteries there is the cranking amps you mentioned. Cranking amps is a peak current delivery specification, but do not try to use the battery at this load for long or it will get quite hot. For this conversation it is the other specification, capacity, rated in amp-hours (Ah) that counts.
A 1Ah battery can deliver 1 amp for one hour before being exhausted. Actually it is a little more complex than this, the actual power delivered will depend on how the battery is constructed and how much current you pull. In general the slower you pull power the longer it will last and it will supply a bit more total power. A 1Ah battery might be able to supply 1.3Ah if you use the power slowly, or only 0.7Ah if you use it hard. The battery manufacturer will have a set of curves that show how this works in a proper battery data sheet.
Back to our example... How many automotive batteries will it take to keep that AC going? We need 840W for 10 hours to meet your specification 840W x 10h = 8400Wh. For this example we will use the common automotive battery, these are inexpensive and commonly available. The average 12V auto battery has about 40Ah of capacity, maybe 50Ah for the larger light truck batteries. This is at 12V, thus 12V x 50Ah = 600Wh. Just one more quick calculation... 8400Wh / 600Wh = 14 batteries to keep the AC going for 10h. This assumes the batteries are all in perfect condition and there are no efficiency losses (both not true) but it gives you a rough number. If building this sort of installation I would put a few more batteries in.
These numbers also do not take into account the DC/AC conversion. That will be done by an inverter and will have an efficiency specification to consider. In conversion you might lose 10-20% of the power due to inefficiency in the process. Add another battery or two.
Twenty or thirty car batteries is something most people do not want to have in their house. Corrosive and potentially explosive gasses, several thousand pounds of weight and a lot of space. The next challenge is charging them, you take another huge efficiency hit, it takes much more power to charge the battery than you can get back out.
When you consider all of the above it becomes apparent why most people using battery or solar/battery power do not use heavy draw appliances (like electric AC, heating, cooking) but use the battery power for lighting, computer and other low draw applications. The couple examples of off-the-grid living I have seen use alternate power (solar water, gas and wood) for those needs.
Using the methods above you can figure out any other situation you want to consider and do the estimates of whether your idea make any sense or not.
