Is it possible to purchase and play a used copy of StarCraft 2?

Do not buy StarCraft II used. You will not be able to play.

StarCraft II is only playable via a Battle.net account with rights to play StarCraft II. There are 2 ways to get those rights:

• Purchase the game online via the Battle.net website.
• Purchase a boxed copy of the game and register the 1-time use key with your account on the Battle.net website

If you do purchase the boxed copy used, the 1-time use key will probably already have been used.

The only thing you could possibly hope to get from purchasing the game used is any of the physical media that comes in the box (the manual, the disc itself). If you haven't purchased the game already, chances are you don't care one whit about any of that stuff anyway.

Edit: @mousey adds that the key has not been used. If that's actually the case, technically you should be able to play it if you do get it. However, it's mostly against the End User License Agreement as stated in the manual:

No Transfer or Sublicense. THE GAME IS LICENSED, NOT SOLD. NEITHER THE GAME NOR THE LICENSE GRANTED IN SECTION 1 MAY BE SUBLICENSED OR TRANSFERRED TO ANY OTHER PERSON OR ENTITY, AND ANY ATTEMPT TO DO SO SHALL BE NULL AND VOID. If a court of competent jurisdiction finds the foregoing sentence to be unenforceable, you agree that you will call Blizzard Customer Service at 1-800-592-5499 to arrange for the transfer of your rights under this Agreement to another person with a valid Battle.net account; provided, however, that Blizzard may charge a processing/handling fee to facilitate the transfer, issue a unique key to the transferee, and remove the preexisting key from the Battle.net account registered to you.account registered to you.

So it sounds like the seller is breaking the rules anyway unless he has actually contacted Blizzard about it. Not that I'm a EULA expert.

Personally, I would buy the game new. But I am incredibly biased because StarCraft II is pretty much my favorite game ever.

You wouldn't be able to transfer a game from an account to another account, since Blizzard doesn't have that function. You can buy the game, but you wont be able to play it since the license key is already used on an account.