Can I Run An Appliance From a 24v Battery?
Just get two car batteries and put them in series.  That should be close enough to 24 V to run the device.  It is quite unlikely that &plusmn;1 V will cause any trouble, although there is no guarantee of that.

Since the power supply is rated for 1.25 A, you know the device won't draw more than that.  Even assuming the full current draw for 8 hours, that comes out to 10 Ah.  You don't want to run car batteries down very far, but even small car batteries can do well more than that.  30 Ah is a "small" car battery, and two of those should be just fine with 10 Ah drained from them.  Overnight should also be plenty of time to charge them back to full.