In this video, the woman says that a sphere is a pretty simple object. What intrigues me is the use of a sphere for such a calculation. First of all, the sphere wouldn't be perfect as a perfect sphere doesn't exist in nature (first source of imprecision). Second, one will have to use π for calculating the volume. As π is irrational, one will have to use an approximation for it (second source imprecision). So why not using a simple cube? The shape would be closer to a cube due it's harder to make a sphere and it's volume would be calculated more precisely. Why they choose to do this calculation using a sphere?
Because spheres are easy to make, easy to polish and they haven't got any edges, which is very much important to create an object of such a high accuracy. In fact, no one said that this is a perfect sphere. It is polished to around a few nanometers away from being perfect. And, there is no macroscopic object that is ideal to a cent percent perfection. So, the first point isn't good...
I suspect that the reason for your confusion is regarding the volume difference between both the case. The volume of a cube is very easy to determine. But, making it to such a high precision is a very big problem (as I've mentioned at first) because of its edges. As David says, right angles are very difficult.
You're right that $\pi$ can only be approximated. Still, we can use the maximum approximations as possible and we would - because, it's gonna be defined under the SI.