How many gallons of water does it take to fill a 24 by 16 oval pool 52 inches deep?

First of all, your 52" pool will have 48" of water. An oval pool is actually an obround consisting of two arcs and a rectangular area between them. The rectangular area is 8 ft x 16 ft, volume of which is 512 cu ft. The two arcs together makes a perfect circle of 16 ft in diameter, and has a volume of 805 cu. ft. Added together and converted to liquid volume is 11,348 U.S. gallons of water.