Yes. It will be possible as soon as Multiplayer is released. How difficult it will be is yet to be seen. But it will be possible.
Wouldn't it cost money to keep a server open?
If you went with a 3rd party who wasn't affiliated with Craig or TM then yes.
There's no guarantee that Craig will provide a service like that but if he did then it would also cost him money so he would be likely to charge for it aswell.
A reliable service with good servers and infrastructure costs a decent amount of money to run.
But if Craig finds a way so people can host it on their computer without the game running and their computer is remained on, then itíll just be electricity bill.
A proper server would still cost less than what it'd cost someone to leave their PC on 24/7 in electric. It would also provide much better infrastructure and connectivity.