Not in the most meaningful sense of "natively and without heavy modifications", because the GBA's ARM lacks the proper MMU. You could always go the route to just emulate something that runs Linux, but it's probably not going to be very usable for anything.

You can configure Linux to run without an MMU - it seems like that should be doable?

Oh I didn't know, indeed: https://www.kernel.org/doc/Documentation/nommu-mmap.txt

Probably not, but there was a neat project from a while back that got an ancient version of UNIX running on it: https://web.archive.org/web/20060831141959/http://www.kernel...

I remember running a version of uCLinux on my GBA SP back in 2006