Perhaps zero, one, more or all of the following:
- duplication of volume seems to duplicate mass and energy, which is impossible
- the Axiom of Choice is false in the real world
- the real world is not based on real numbers (ironic)
- mathematical measures are not related to real world measure(ment)s
- there are no realizable infinities or infinitesimals
- the real world is ultimately discrete at the lowest level
Important to the paradox is that it does not present a construction of the division. And more importantly, it relies on the fact that it is not possible to construct such a division.
If you accept some notion of the Universal Church-Turing Thesis as a matter of reality, then it is not possible even in principle for such a division to occur.
Note that this is not just a matter of "finding the right way to construct it" -- the theorem critically relies on the fact that given any way of constructing sets using finite representations (like computer programs or layout of molecules), the theorem does not hold.
The 'axiom of choice' isnt a property of the real world one way or another.
The issue is that physical measures (length, area, volume, etc.) require measure theory in mathematics to specify properly. The issue is the mathematics was too simple to model the relevant physical systems.
All of physics is specified in terms of a spacetime continuum, neither this, nor anything else, indicates that physics requires a revision.