Is it possible to host a game with a website as the host?
You can set up a permanent server (not the same as a website), but Dream Daemon must be installed on it. You might be able to get an ISP to provide this service to you, if you're lucky (and preferably wealthy...).

If you meant webpages made using BYOND (like Bwiki does), check out the CGI library.

There are, however, a couple catch-22s there: if the server has the BYOND software installed, then you can use DMBs that were compiled in CGI mode as web scripts. (For example, ByondBwicki is a DMB-based script.)

To be able to host an actual persistent (or semi-persistent) game from a server, you need to have a "shell account". Those are usually extremely expensive, and most shell accounts are for UNIX-based servers only.

You're probably better off getting a high-speed modem, figuring out how to open a hole in the firewall, and hosting the game yourself. Either that, or finding a (good) friend who will host it for you for free without owing any favours.

(Remember, all that a host needs is the .rsc file and the .dmb file. The only other files they would need are any special data savefiles you've generated for the game, if your game is designed that way. Anyone saying they need anything more is outright lying, whether through ignorance or deceit.)