It seems that a user by the name of Ryuta has managed to get StarCraft up and running within a browser. The files needed to get the game up and running has since been uploaded onto GitHub so if you’re so inclined, you could actually get the game hosted on your own server and start playing it.
However it should be noted that this isn’t the full game per se. Rather it’s a series of demos that players can take for a spin just to see what the game is like, and we suppose to a certain extent show off the power of the HTML5 standard which many believe will soon takeover Flash completely.
That being said if you don’t want to host the files yourself, someone else has done it already and you can check it out here. However we expect that given Blizzard’s fierce protectiveness over their IP that we don’t think the hosted version will be up for long, so you’ll want to take it for a spin while it is still online.