Thanks for your thoughts!
Still kind of on the fence on whether to use old or new hardware, but one idea I was thinking about is that the MinnowBoard might serve as a good base. It should have fast boot, and enough power to run Dosbox comfortably, and it's tiny. Probably uses wayyy less energy than old hardware, too.
Of course there's also the Arduino Due/Raspberry/BeagleBone, but I'm not sure if they're powerful enough (Dosbox performs the best on x86, right?)