Just wondering, are there any projects like WINE, but for Mac software? That is, can we use a program that acts like a compatibility layer to run Mac apps?
It appears to be developed for academic purposes and is still in its early stages, but it looks promising. It is based on the aforementioned GNUstep.
There was a project attempting to do this for very old Mac Software (68k) called Executor: http://en.wikipedia.org/wiki/Executor_%28software%29 -- it is, of course, defunct.
There is no modern equivalent as far as I can tell, and making such a thing would be almost as big a task as Wine itself.
EDIT: Reading the other answers, it appears that there are now ways to do what OP asked. I'm leaving this answer here because it is still a viable alternative on some hardware for applications which won't run under emulation.
While the other answers are correct (no, you cannot), there is a way to install OSX on Virtualisation software such as VirtualBox. This is not what you asked for, so be careful, but it may get you the results you need.
You cannot run Mac applications on an Ubuntu machine. Even if it were possible, it would be very hard to satisfy the requirements of most applications.
Most Mac applications use features like
/Library/Application Support, something Ubuntu doesn't have (to my knowledge). Some apps even modify system files or add new ones (The former could break your system if they happen to be in the same location).
If you really want to run Mac apps, your best bet is to either virtualize macOS or boot it natively. The latter would (most likely) be the hardest; it's called hackintoshing and it's a very popular method (but breaks the Apple EULA).
If you can find a distro of macOS that would work on your hardware, it's not a bad idea to give it a try if you really want native support. However, it could be a tough job. I won't go into much detail, but most of what you'd normally want to know can be found with a quick Google search.