Recently I came across this too when I needed to diff two large files (>5Gb each).
I tried to use 'diff' with different options, but even the --speed-large-files had no effect. Other methods like splitting the files into smaller ones, using xdelta or sorting the files as per this suggestion didn't help either. I even got my hands around a very powerful VM (> 72Gb RAM), but still got this memory exhausted error.
I finally got to work by adding the following parameter to sysctl.conf (sudo vim /etc/sysctl.conf):
vm.overcommit_memory=1
vm.overcommit_memory has three values (0,1,2) and sets the kernel virtual memory accounting mode. From the proc(5) man page:
0: heuristic overcommit (this is the default)
1: always overcommit, never check
2: always check, never overcommit
To make sure that the parameter is indeed applied you can run
sudo sysctl -p
Don't forget to change this parameter back when you finish!