Can I still play Starcraft 1 on Battle.Net?

Yes, you can. You still have to have a valid SC1 account though. And it uses Classic Battle.Net, so you can't hook up with RealID friends from SC2 and WoW.

Blizzard has released Starcraft 1 for free. If you download that version (you'll probably need to login into your blizzard account in order to do so), you'll be allowed to login into battle net without any problems.