Paying helps, but even then, they are usually terms against abusive behavior. It seems that what constitutes abuse reduces as you go up the chain (for example: perhaps a social network forbids false information and worse, perhaps a hosted CMS forbids hate speech and worse, perhaps a VPS/colo/residential forbids only very illegal activity) but it's always there to some extent. Maybe it's about human abuse and maybe it's about technical/network abuse, but there is something forbidden.

Is it possible (not feasible, but possible) to put something on the internet that relies on nobody else at all?

Sure. Just invent your own Internet.

Is this actually a valid (albeit not feasible, but theoretically valid) answer to the question though? I didn't capitalize the I in my question, but I should have, because I do mean for it to be interoperable / peered.

I think so? If you run your radio transmitter. Just tell peers what frequency. Though if some company decides to build a large relay node infrastructure, you'd need some crypto verification to keep them off Yournet.