You need a T1 at least or similar connection. (Same downstream with upstream.) I once ran a service on 2 cable modems with an interesting yet legal hack – but doubt you could do that today.
You need to make sure the parts/hardware is stable.
You will need a stable OS for hosting (I’ve always used linux or freebsd) – depends on your preference.
It is a good idea to have a user control panel for them to do most of the configuration of their account themselves, also to manage accounts and what not (billing etc) – you can look into http://www.vhcs.net/index.html – or commercial solutions – cpanel, plex etc.
Find a list of common services and configurations you need to provide – just looking at any webhost like http://bit.ly/a5zzEB (They are one of the reasons I’m not into hosting anymore ;)
What is absolutely essential is having a daily (or less) backup system. With retention of at least a month. I used to use tapes. Daily tape, weekly tape, monthly tape etc. See what’s available right now. Make sure you know how to recover things from backups etc.
Always keep everything up to date. There’s plenty of attack scripts out there running non-stop ;) (Got to love script-kiddies etc)
I assume you’re doing this for fun and learning, or personal hosting/friends family stuff like that.