TLDR; they do exist (like ZeroNet, which I don't recommend), but they always fail, because viruses and illegal porn are everywhere.

Like qbittorrent but for chatting instead of stealing game of thrones.

I laughed at this. It's mostly used for pirating, but does have some legal applications, like downloading large files or Linux distros — which is what I use it for, since some distros are quite large, and torrenting is often faster than downloading. That's how I got Linux Lite for my mom's computer.

Is it technically impossible

It would just be moving files around. You'd have to make sure only files you need are being downloaded, though, so it doesn't slow down everyone's devices and fill up their hard-drives.

or technically a pain in the ass

It would be almost impossible to prevent viruses, malware, and illegal porn from spreading like crazy. That's what happened to ZeroNet, which I stopped using once it downloaded a crypto miner to my computer.

or there's no way to make money off it

There wouldn't be a need to, since content would be distributed, rather than centralized in a single server. I guess you could make money off of it with donation links, but that would be like spamming ads.

nobody wants to give up being servermaster of mount internet

That's the major problem. The big tech cartel makes it hard to create, find, and use alternatives.

would you necessarily have to expose your IP to everyone else

Yes, but you could use a VPN or proxy. They just need to know where to send the files, and your IP address tells them that. This does open the door to deanonymization, though, especially with malicious files — which is, quite honestly, the biggest hurdle so'm like this faces.

it'd just be a big dox machine?

Contrary to popular belief, IP addresses don't give you very much information. You can plug it into a website to figure out where someone lives, but this is only a rough estimate. When I plug mine in: it always says I live in a town I've never even been to.

The most important information it can give is your ISP. This ain't useful for most people, but the government can demand your ISP to reveal who used that IP address at a certain time to figure out who you are.

IP tracking is probably the largest threat — and mainstream social media already does this. They track which IP addresses visit their website, log what they do, and try to get records from other websites. In this case: these companies could just monitor the network.

There's also the threat of hackers targeting you, but file sharing is a bigger problem on that front, although someone might try to DDoS you by making a bunch of fake requests.

But, no, it wouldn't reveal who you are to anyone who can't already figure it out.

So far the best anyone's done is federation

Well, no, distributed platforms like ZeroNet do exist, but they're inherently impractical due to the prevalence of viruses and illegal porn.

So to answer your question: there are way too many viruses and illegal stuff to make it practical. The risk is way too high unless you know the people you're talking to — and even if you do: what if they get hacked? Or what if they just get impersonated?

It ain't gon'o happen. ZeroNet, and plenty of other platforms, tried this and failed.