The Sega Dreamcast has a processor, ROM drive and built-in modem. What’s stopping somebody from hacking whatever OS it uses and programming it to act as a web server?
And it is going to store the information where?
Dreamcast has to have some kind of cache memory where it holds texture maps and geometry it’s read off the CD ROM. Or maybe ever on the CD ROM itself.
I don’t see any reason why not. You can already run linux on it, if you’re clever enough. I saw screenshots a while back of some guy running xmame on his dreamcast.
On the other hand, not having a hard drive would suck a little.
Okay, a Google search finds that loading Linux onto a Dreamcast is quite do-able. The question is what have you gained? Sure there is a cache of what 16 megs? And how do you make changes to content? Yes, it could be done via the CD-ROM, but you would still need a computer to make the actual changes and then burn the CD. So, you also have a modem, how many active connections can you have? Probably not enough. Also, how would visitors access the information? If you have, lets say 5 connections, how are they going to hit the CD at the same time?
Can it be done, most likely it can. The real question is why would you want to?
NetBSD has a Dreamcast port. IIRC, this port is officially supported by Sega.
Because you can? I know of a few Commodore 64 web servers out there for no other apparent reason other than to prove that it can be done.
You might want the broadband adapter.
As for why you’d want to do it… if you aren’t squeezing every last bit of performance out of your electronics, you just aren’t getting your money’s worth. People give my roommates funny looks when we play NES games on the Dreamcast, but that doesn’t stop us. 
Honestly, the processor is hella fast for the price, it doesn’t draw much power and since it uses a consistent hardware configuration, it will likely be rather stable. If it’s a web site where the content doesn’t change much and you’ve got the adapter, then why not?
Besides, it’s much more l33t than using a pc. 
What a totally crazy concept. I just want to say that the fact that the answer is “yes” completely blows me away.
It still makes no sense to me, but if you want to try, this is a good place to start.