Is there an internet Git-like repository for collaboration on a paper?

Aditya's comment should be the accepted answer:

Bitbucket offers unlimited private repos for academic users.

Overleaf (used to be writelatex) now works with git. Latex is fantastic for typesetting academic papers properly, and Overleaf is great for writing latex collaboratively.

Consider asking your research institute/university IT services.

• In terms of fees, I'd consider such a tool as necessary for scientific work as your office chair, or backup disks/servers. My personal experience with asking for a git repository on a file server was that our director immediately answered "if that's what is needed for work, it needs to be installed." - that was it.

• The far more important concern is, where research in progress and possibly data will end up. Storing such sensitive data outside the collaborating institutes is by default a big NO, even though lots of people send their research data by skype or dropbox.
  You'd be totally screwed up if you run into ownership/privacy problems.
  Even Bitbucket (who have "academic license") have write in their terms such statements (there are other statements about you retaining ownership of your data, privacy etc.):

  End User hereby grants Atlassian a non-exclusive license to copy, distribute, perform, display, store, modify, and otherwise use End User Data in connection with operating the Hosted Services.

• For some areas of research it is even more NO, e.g. I work with patient data...

• git works very well with distributed systems, even if they are only seldom connected. Worst case, people can email patches.

• But installing git on a server is easy and people may get an ssh login, and that is all you need to have for your private git repository inside your institution IT structure, which avoids all that privacy trouble.