If git is failing due to a system-wide file limit, that would explain why other programs are also failing. Perhaps some process has gone mad with creating files and if you find and delete those it will help things recover. (I agree with your general point -- just trying to be helpful in getting your computer back working.)

Google "too many open files" and you'll find posts about OS X but also Linux. Could it be git itself that's hitting a limit? Maybe you're hitting the shell's ulimit, not the system's per process or per-system open files max.