From what I can tell, the oldest running web server that is active and online today is surprisingly a Commodore 64. The specs on this machine are as follows: 64k Ram, 8bit 6510 MOS-Technology CPU with a clock speed of 1Mhz. It is claimed that this machine is also unmodified.
Earlier today I was able to visit the main website hosted on the web server, www.c64web.com. It loaded slow, but in my opinion it performed amazingly well knowing that it was being served by a Commodore 64! I don't know what you think, but I think that's just amazing.
Unfortunately though, since my one brief visit to the site, I haven't been able to load it back up again - Google has a cached copy though. I am sure it will be back online again soon. One thing for sure, after discovering that a Commodore 64 can perform as a web server, I don't have any problems continuing to host my own websites from home with the old equipment I use.
Does seeing this inspire you a little more to take matter into your own hands and start hosting your sites off your own server? You can literally dig out any computer from the trash, clean it up, install Linux, Apache, PHP, Perl and MySQL and it will fire right up and start serving pages to anyone who cares to view them. It really doesn't take a lot of computing power to serve web pages. I think discovering the Commodore 64 web server supports this.