Short history of Nauru, Australia's dumping ground for refugees
Australians demand end to Nauru refugee centres
RIP New Zealand
It was a good run
They can't be born there because the citizenship is conferred to men only*, and that only for a limited time (unless you're the Pope), i.e. ius offici.
This means you can only get Vatican citizenship if the Pope wants you to work there, basically - and you only hold that citizenship during your stay there.
*Women married to men working there sometimes can also get citizenship if the Pope feels like it during their stay, but that's not guaranteed. Since most men permanently living there are clergy of some sort and have to be celibate anyway, this rarely occurs.