Is LAN with StarCraft 2 definitely not possible?

Judging from your question you want SC2 over LAN without connecting via the Internet. There is no easy way to do this. As McKay points out, as long as all computers are connected to the Internet you can use computers which happen to be connected to the same LAN, but make no mistake you will not be leveraging the advantage of said LAN.

Now I said "easy way," because there is a "hard way." Blizzard does have a LAN client of Starcraft II which is provides for reputable tournaments. Gaining access to this requires setting yourself up as a business organization and paying a licensing fee. For more information you're going to have to contact Blizzard directly.

That said, for a couple of friends, there is no legal way to leverage your LAN for SC2 games without going through Blizzard.