It's not really a server but you can do some interesting stuff with intercepting fetches in Service Workers, e.g. https://servefolder.dev - a little side project of mine.

It depends on the definition of "functioning".

If you'd like to access a server from another tab / iframe of the same browser, that's almost possible already, just some UX work would be required.

If you'd like users of the same page (or a separate specialized page) to connect, that could be possible with WebRTC.

If you'd like arbitrary hosts to connect, that would require a server side proxy, there is no client-side only solution that I can see,

I have it working here https://webcode.run

Use Opera?

https://www.ctrl.blog/entry/opera-unite.html