Last year as a router for my home network, I dug out a macintosh quadra 700 to be the server. I always knew it was old, but it suprised me that it was 10 years old and ran so well (300k a sec on a cable modem) I know you can get a mac plus on the internet (1985), what about a IBM AT or XT? Would it be possible to get an origional VAX on the net still? What about old sun workstations. I have 3 weeks off and need a project.
I am sure I am not a thechno-saavy as you, but I have a quick question: Are you trying to view text only, like many of the smaller hand-helds(i.e. Blackberry, Palm VII, etc.) or are you attempting to view images as well? My guess would be that you are limiting your options(as far as how deep you can dig into your garage 
) quite a bit when you attempt to view images.
The oldest I have seen is the Commodore 64!
It should probably not surprise you that there are still old fogeys out there accessing the net on dumb terminals. I could probably get onto a shell account at my ISP with my old vintage 1975 Sol Terminal Computer.
But a more useful question is if there are any real antique machines still providing services to the net. We tossed this around on Usenet in alt.folklore.computers. Several people reported old VAX (i.e. VAX 11/750) but these seem to be operated by computer hobbyists. For the oldest continuously online machine, it appears that there is one PDP-11/70 that’s been in continuous service in Sweden since1975. Other units like the PDP11-23 are also reported to be still in service. These DEC PDP11 machines were about the first internet-capable minis so you’re not likely to find anything earlier that’s still in operation. I’m sure there are more old machines out there…
I personally have two ten year old macs that I occasionally use to surf the 'net. If you’re just talking about using a dumb terminal, I could probably hook my apple iic and telnet in to a lynx session. That would be cheating, though.
The oldest computer I’ve heard of using as an actual web host was a 386 with 20 mb of ram. The creator actually set out to make the smallest possible web server on a PC.
I have a feeling you might be able to get an even older Mac, say a MacII to work as a host–mac hosting software is more forgiving than PC.
I knew of a text-only web site that was hosted on an unaccelerated Mac SE, using…what was the name…some silly name…“MacHTTP”? “HTTP’d”? Older than WebStar. Anyway, an 8 MHz machine with 4 MB of RAM, built circa 1986 or 1987. Nearly 15 year old technology. I’m sure you could go a lot older if it doesn’t have to be a personal-computer type of box, though.