I tackled this in College back in the day. So I believe the significance of the problem is that you cannot use a straight edge and compass to solve this problem. I don’t remember if it is because of the dimensions or not. But basically, as it was schooled into me by my awesome professor…
A cube is a box in three dimensions where the edge length of X is equal for all edge lengths (read not a rectangular cube).
If you have an edge length of X = 1, then the volume of your cube will equal 1 because X^3 = 1^3 = 1. If you want to double the cube, it is not as simple as doubling each edge length. So you need to find a value such the X^3 = 2 essentially.
To find the new value of X, lets call it (Xx) you need to find the third root of 2 or as the problem states, the cubed root of 2. So Xx =3rdRT(2) (I don’t know how to do this outside of Wolfram).
So with a little inference and proofing, Xx would need to equal a value of r, where r is a construct-able value. In Geometry, that means Xn needs to be a real number that is construct-able with a straight edge and/or compass in a finite number of steps. Thus Xx must equalt |r|, which must be real.
In Algebra, r can only be construct-able IFF it can be constructed with values of integers 0 and 1 and only using Addition, Subtraction, Multiplication, Division, and SquareRoots only.
Taking what we have just established, we can say that Xx needs to be equal to |r|, which must be real, which means Xn must also be rational. So working backwards, Discreet numbers are integers. Integers are rational numbers, and thus Rational numbers are real numbers (R(Q(Z(N)))) (where N are natural numbers).
All of that to say is that since we are trying to find the Value of Xn such that the cube with volume 1 will be doubled is impossible, because the smallest polynomial that will get you the volume of 2 is of the 3rd degree, making the value of Xn a non integer, which means that it does not satisfy the requirements to be a Real number, thus it cannot be a real number and thus is not rational and would never satisfy a value of |r| such that it can be geometrically construct-able using a compass and straight edge in a finite number of steps.
Did that make sense. Been out of college for a little bit but I remember most of this because it was a week long assignment, only to find out that our professor put us on a fools errand. Also, my mathematical proofs may be a little dusty as I don’t really do this in the IT space anymore.
**Edit
I was working out the proof in real time as I type it out. So yes, how ever you want to look at it, the easiest answer is that due to the dimension of the polynomial, you cannot square the cube and keep it construct-able. So yea, the dimensions are probably the biggest significance of why the problem exists.