How do I change the video settings from outside Starcraft 2?

Under My Documents/Starcraft II in Windows or ~/Library/Application Support/Blizzard/StarCraft II in Mac OSX (not the Starcraft II installation folder), there is a text file called Variables.txt which contains all the basic settings for the game, including graphics and sound options.

In this file, the variables height and width determine respectively the height and width in pixels of your game window on launch. Changing these from (in my case) 1080 and 1920 to 800 and 1024 fixed the problem.

Pressing alt+enter (Windows) or ⌘+M (Mac OS X) should make the game run in windowed mode. The key-combination should usually work as soon as the login dialog appears, so you might have to wait a short period of time.

Once the game runs in windowed mode, you should be able to go to the settings and change your display resolution, then switch back to fullscreen.