Is it possible to configure a discourse site to completely disable direct / private messages?
Yes, see the
enable personal messages site setting and also something more specific to users Allow users to disable new PMs
Sorry, I was not being clear.
I am not looking for a way to allow individual users to disable PMs.
I want a way to, as site configurator / admin / root, to globally disable PMs for everyone.
Yup! I mentioned that there too.