It is The Darling Project that's set out to achieve binary compatible support for Apple OS X / Darwin applications on Linux. While in its early stages, Darling does leverage some code from GNUstep, an open-source implementation of Apple's Cocoa Objective-C libraries, widget tool-kit, and application tools.
Darling also possesses some hope as the project is being worked on for a diploma thesis research project by a university student, so at least this means the code should continue to be developed for several months to come in a serious manner. This work is being done principally by Luboš Doležel.
Darling works in a similar manner to Wine for Windows programs but instead it's OS X binaries. Darling must provide an ABI-compatible set of libraries/frameworks as those on OS X so it can parse the executable files for the Darwin kernel, load them into memory, and execute them without needing any code recompilation or other modifications for Linux. Darling uses the GNUstep libobjc2 run-time for part of this Objective-C support. While not a focus of the current work, it's also theoretically possible that it could be extended to handle running applications for Apple's iOS, but that would require dealing with ARM and supporting additional Apple frameworks. However, Luboš Doležel doesn't seem too interested in taking up this iOS on Linux challenge.
Among the components making up Darling right now is a Mach-O dynamic loader, a tool for Mach-O file examination, a Mac binary file extractor, and a set of libraries. The libraries provide Mach file loading support, a Darwin libc/libSystem replacement, an Apple Objective-C runtime translation layer, and extractors for various formats used on Mac OS X.
Unfortunately at this time, with the project still being early in development, there aren't many Apple programs that can yet run on Linux with this translation layer. Most of the OS X applications working with Darling at this point are console programs. For those interested in the Darling project to run Apple OS X binaries natively on Linux, see the Darling Project Wiki.