Could this be achieved? It would have to be done legally, without infringing on Apple's copyrights.
Could it be built on a Darwin, FreeBSD or other base and then aim for binary compatibility with OS X software? Similarly to what React OS attempted, but perhaps with a greater chance of success?
Edit: The difference between this and the suggested duplicate is that the other question is about a particular software that already exists for existing operating systems (Linuxes). This question, on the other hand, is about the feasibility of building a whole new OS (possibly using a Darwin or FreeBSD base.)