How do you mean, run?
Compile the kernel so that the processor could run it? Sure. I'd be surprised if the chip isn't readily supported, it's just ARM after all.
But compiling the kernel and having the eMate run it with its 3MB RAM would be tricky, and how would you load it? You'd have to make new ROMs.
That's before the issue of device drivers come into play. I don't think they're available... and I'm quite sure the hardware isn't documented in any way outside of Apple.
If what you're asking is if there's a Linux distribution for the eMate, then no. I'm quite sure there isn't.
You can use it as a terminal, though, to get a serial connection to a bigger box running Linux. I'd like one for that exact purpose (though I don't actually run Linux, as I prefer BSD).