That's fine if your service is really local - you can do it at night. Not really an option for a global site. Imagine if Twitter went down for a few hours every month. People are addicted to Twitter. It might be at a critical time for an entire country (e.g. the Queen dies). Even worse you can't guarantee how long the upgrade will take.

You'd definitely need at least two servers. But I think you could surely just have simple master/slave replication and switch between them.