Probably not too popular of an opinion on HN but email in my opinion would be a great example of a service that could be run by the government. Just like postal service (at least in some parts of the world)

There was something like that in Germany called de-mail. It was official and receiving and reading a mail was considered legally binding (invoices, etc.)

It could have been great but the implementation lacked encryption and had wild security issues. So nobody used it and it was shut down