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