4,096 cm^3
it's useful to know that the box with the largest volume - to - surface area ratio is a cube. Your shape will be a cube (with a missing top).
Thus there will be five square faces with a total area of 1300 square centimeters. The area of each side will thus be:
1300/5 = 260
The length of each side will again be an irrational number (√260), because 260 is not a perfect square. This comes out to roughly:
16
The simplest way to do it from here would be to simply take the cube of your side length:
163 = 4096
Your box's volume will be roughly 4096 cm^3