Two different Omeka sites

I already have an existing site, but would like to create an entirely new archive (different items, collections, etc) completely unrelated to the existing one. Would this require me to download and install a second version of Omeka in a new directory, or is there a way to separate the two sites that I'm not thinking of?

Yes, you will need to install a new version of Omeka in a new directory, with its own database (or extension of an existing db).