How to close a file in Intellij using a keyboard shortcut?

hmm you can't because there is no default key binding for that, at least on OSX, however you can go here and define one:

Idiot me. CMD-F4 or, as mentioned before, CTRL-F4.

Thank you all. I'll just bury myself now :)

The standard tip if you don't know how to do something in Idea is "Find Action..." (ctrl+shift+A in Windows) then type what you want to do, in this case close.

You will get a lot of different "close" actions and simply "Close" (ctrl+F4 in Windows) will be one of them.