You would have to burn a cart that contained the webpage and TCP stack, as well as having the ethernet hardware, but there are tricks you can do to reduce the memory requirements. It's not going to be able to handle more than a packet or two at a time, but I've seen TCP stacks squeezed onto really low end hardware. Obviously you won't be able to open much of a TCP window so performance will be lousy, but it's a 1.8Mhz CPU so that was always going to be the case. Just don't have any misconceptions about being able to run TLS on it. Remember that TCP was developed on machines that did not have a lot of memory or even CPU cycles.

An Atari 2600 might be a bridge too far, but a NES should be able to do it.

Fair enough, seems a lot simpler than what I had in mind with expansions and custom software, but also a far cry from OP's "flash" a general purpose OS and more or less get to working

You would be surprised with what a Jupiter Ace could do with few KB of RAM. Add a external module for serial <> PPP and the fun begins.

Funny how that works. It's almost impossible on fifth gen but trivial on sixth gen (at least on the Xbox, which pretty much was a PC).

All the mainstream 6th gen consoles had first party support for wired ethernet; although only the Xbox had it built in (well, the slim ps2 had it built in too). I'd expect they've all served web pages at one time or another.

5th generation could probably make it happen, but without ethernet, you're looking at a modem or serial interface, and in today's world that's almost certainly talking to something else in your house that's a better host. :P Wikipedia categorizes the Apple Pippin as 5th gen, and it's more or less a Mac with a weird pinout for the PCI slot, so I'd guess it's the easiest to get ethernet with; without resorting to something that has more smarts than the 'host'

Even with Ethernet, aren't you're plugging the console into a switch that can probably serve webpages as well as the console?

Most of my switches are unmanaged, so probably not. Maybe the dsl or cable modem or whatever can serve pages, but especially if it's isp owned, it might not be easy to.