Battery sizing for motor?
This sounds like a classic word problem where the trick is to know which 80% of the information to throw out.

The rest is contradictory.  You say the load requires 24 V at 11.5 A.  That comes out to 276 W.  But then you specify 180 W.  Huh?  Which is it?

Let's say 11.5 A is the stall current, but 180 W is the running power.  The startup will be short relative to the 20 minutes run time, so this comes down to needing to supply 24 V at (180 W)/(24 V) = 7.5 A for 20 minutes.  The maximum current output needs to be at least 11.5 A.

The load requires 24 V, so that should be the battery voltage.  7.5 A for 20 minutes comes out to 2.5 Ah.  Let's say you want the battery to run down to no less than half full, so you need at least a 5 Ah battery.

Two 12 V sealed lead-acid batteries of 5 Ah capacity or more would do fine.  For simplicity, you could use two 12 V car batteries in series.  You won't be able to find a car battery as small as 5 Ah, so any two of the same type of 12 V car batteries in series will do.