It's a little-known fact that the Vatican has its own train station; it is rarely ever used. Historically, it has been used by the Pope for special travel on Italian rail or to send off papal remains. By special arrangement with the Italy State railway, ordinary people (usually train enthusiasts) can arrange to arrive and leave the Vatican via the Vatican Station. This requires quite some time to arrange and is not remotely cheap.
Consider me interested. However Wikivoyage is lacking info on how to achieve this? Can anyone else provide some information on this?