Apparently there's a bug that prevents players from creating a Hardcore Mode character. There was no character wipe in this beta patch, which seems to cause problems. Hardcore Mode (beta only) is unlocked at level 10, but having existing characters beyond level 10 prevents the game from creating a hardcore character.
I just leveled a Wizard from level 1-10, and the Hardcore mode button was unlocked, but I'm still unable to create a Hardcore mode character.
Blizzard added this bug into the Known issues forum thread.