Technically you can store data on an external hard disk. But prudence dictates that you need a second external as backup device. There are FAR too many posts in this forum of failed externals to use them as even a moderately safe place to store files.

An external disk, moreover, is much slower than in internal one. So running programs off such a disk (and those huge game files are in the same folder as the executables) definitely is a bad idea.

The best thing to do:
1. Buy and install a second internal disk. All desktops can accomodate it.
2. Buy an external disk for backup purposes.
Do BOTH!

Kees