If you want to construct a box with a volume of 52 cubic feet then, as you said, you need the length (L), width (W) and height (H) to satisfy
L W H = 52
You are free to choose any numbers that work, for example
13 2 2 = 52
so you can make a box that is 13 feet long, 2 feet wide and 2 feet deep. Likewise
26 2 1 = 52
so you can build a box that is 26 feet long, 2 feet wide and 1 foot deep.
But you seem to want a box where the length, width and height are all the same measurement, so that the box is a cube. In that case you want a number x so that
x x x = x3 = 52
You tried x = 3, x = 3.5 and x = 3.75 feet. This is a perfectly good way to solve this problem, you have a sequence of approximations that are approaching the correct answer, and you stop when you are close enough. In your example 3.753 = 52.73 which is a little too large (you are out by 0.73 cu ft) so you might try 52.73 and get
3.733 = 51.90
which is a little too small, but only out by 0.1 cu ft.
Is there an easier way? That is, is there a way to find x if x3 = 52? If there is a "cube root" button on your calculator ( 3√ ) you can use it, otherwise your method is as good a any other. My calculator gave the cube root of 52 cubic feet as 3.7325 feet.
I hope this helps,