The source code of the game engine i.e. iDtech 3, is free. But not Quake 3 Arena, the game.
However there are free games based on the same engine i.e Open Arena & Alien Arena. They are based on the same iDtech2/3 and have similar deathmatch style gameplay of Quake 3.
Edit: Alien Arena uses idtech 2 not idtech 3. Sorry.
The game is not free because, the char models and maps are not free. But if someone removes all the copyrighted material from the game and programs to use some other models and maps its free. openarena for linux is free. But the model details is not good.