From OdaWiki

Odamex supports a quake-style console system. You can access it by using the tilde (~) or grave (¬) key, depending on your keyboard layout.

If this does not work, use the 'Escape' key to open the menu, then by using the arrow keys and Enter: select "Options", then "Customize controls", then "Console". Hit a key you wish to use to access the console.