Is there a git-like file system?

What you're looking for is called "deduplication". While it's usually implemented by vendors of specialized storage products, the ZFS filesystem implements it as well. Most Unix-derived operating systems can make use of ZFS, and I'd therefore recommend it as the first place to look.

Network Appliance, Inc. has had storage ability like this for many years, in fact they had complaints filed with Sun Microsystems for their ZFS file system, and does what Aaron Miller is mentioning in his accepted answer. For what it's worth, the complaints were settled with Oracle after about 3 years.

I have used this as a corporate solution since 2000, and it works well. The cost of storage after the first 'copy' comes about once there is a change in the file. Otherwise, with many 'copies' comes only a slight increase for namespace.

I don't think this answers the question about "operating system level" but rather on a "file system level".