This is possible obvious, but how do I get Doom 2 to run in 120x1080, rather than a full screen black box with the game a small rectangle in the middle?

Guessing that you are running this via DOSBox, using the game "out of the box" from the GOG installer. Personally I'd use a source port (I use GZDoom, but there are others available, like Zandronum and Chocolate Doom).

However, if you want to use DOSBox as per the GOG installation then GOG usually provide a DOSBox configurator to easily change some of the settings it uses, so check the options there. There's probably a GOG provided shortcut to the configurator but if you can't find it then just run the executable from the folder where you installed the game.

If this still doesn't help then you can modify the conf file that DOSBox uses. You'll need to look at setting the below settings (assuming that your monitor is running in 1920x1080):

fullscreen=true
fullresolution=desktop
output=
aspect=
scaler=

I've left the last three settings null here as you may need to adjust them to suit. I'd probably start off with output as ddraw or opengl so that your graphics hardware gets used. Aspect can be true or false, and whether it makes a difference may depend on any graphics driver settings you have in place and other DOSBox settings, so just try out both ways to see which you prefer. Lastly the scaler is just something to adjust the image, like getting rid of jagged edges by upscaling the image and smoothing it. There are a variety of options, but for no scaler just use scaler=none and then it will keep the pixelly appearance.