The Wine project lets you run Windows programs on other operating systems, such as Linux.  I spent some time recently trying to see what it would take to run Visual C++ 2013 Update 4 under Linux using Wine.

The first thing that I tried to do was to run the installer, but that unfortunately hits a bug and doesn't work.  After spending some time looking into other solutions, I came up with a relatively decent solution which seems to work very well.  I put the instructions up on github if you're interested, but the gist is that I used a Chromium depot_tools script to extract the necessary files for the toolchain and the Windows SDK, which you can copy to a Linux machine and with some DLL loading hackery you will get a working toolchain.  (Note that I didn't try to run the IDE, and I strongly suspect that will not work out of the box.)

This should be the entire toolchain that is necessary to build Firefox for Windows under Linux.  I already have some local hacks which help us get past the configure script, hopefully this will enable us to experiment with using Linux to be able to build Firefox for Windows more efficiently.  But there is of course a lot of work yet to be done.