Is there any way to access old YouTube videos which have been rendered unavailable if you still have the URL?

I thought maybe archive.org might keep a copy of the page up, but I suppose that would be far too much data to store and YouTube would certainly take them down.

It's for something really stupid at the moment, Limmy playing an old computer game, but if it can be done I thought it might be a useful skill to know in the future as well.