Probably not, that's why there are web hosting companies and game server hosting companies. Most web hosting servers restrict you from doing this sort of thing. I know mine did back when I was with Bluehost.
Rigging up your own server is a much easier prospect, but unless you have a really big-ass connection and a dedicated computer just for hosting the server, it's gonna lag like a sonovabitch.
If you're looking to host a GoldSource (or Source too, I guess) game, get the dedicated server update tool from the bottom right of
this page and follow the setup instructions detailed on the Steam forums. That's what I did.