A domain is a yearly expense.

I'm not sure what you mean by "Self proclaim a domain"...

Basically, domain points to server, server does everything else...

Now, if you just want to host a site, you don't need to own your own server, or rent one, a standard webhost would work, however, you'll only get what they want you to have (as in whatever they install you're stuck with, as it's not a dedicated or virtually dedicated machine)