22amps at 3 hours is 66 amp-hours. There are batteries that can do that, they're large car batteries. However, generally the amp-hour rating of batteries is for the 24-hour discharge rate (as in they can supply 66 amps over a period of 24 hours, rather than in one small chunk...this has to do with internal resistance of the battery). If you expect to do this on a regular basis, you'll definitely want a deep-cycle battery which is designed to be discharged fairly far and then recharged.
Personally, I think you're going the wrong route and should look into a system that consumes much less power. With your system, you're looking at an expensive battery that you'll likely destroy in a few months with discharge/recharge cycles like that.