I really wish I didn't have to post on this accursed forum, but all searching this issue didn't turn up anything useful, and since it's the GOG version, you... PEOPLE might be my best chance at fixing it.
Elder Scrolls: Arena has no sound on Linux. I specify on Linux because I've tried both my own installation of DOSBox as well as GOG's included DOSBox version through WINE. Neither provided any sound. Changing the settings in the included INSTALL.EXE didn't fix it, nor did altering DOSBox's MIDI settings. None of the other DOS games I have on my system had this issue, the closest being CD audio in HeXen and its expansion, an issue which was easily fixed by moving the imgmount command in the conf file to after c:.

GOG Forums. Useless as ever.
Anyway, I figured it out. Arena and Daggerfall both use General MIDI, which requires you have fluidsynth or whatever MIDI playback software set up as a MIDI device in the game's config files (Run pmidi -l in terminal to figure out what port fluidsynth's using and write it in "mididevice=") and running in the background. If you plan on using a shell script to launch the game like a goddamn normal person, you'll have to run it as "(fluidsynth command) & (game command)" or it'll only launch fluidsynth and not DOSBox.
Thanks for nothing, losers.
Edit: You don't need to do this for Battlespire because that uses CD audio, but you might need to move the imgmount command to after c: if you're experiencing crashes playing that.