Does a domain name matter?

In thinking about search engine optimization, search engine indexing, and the price of web hosting, I can't help but wonder about the impact of domain names, IP numbers, and whois registration information.

For example, does this blog, which gets published to Verizon's "mypages" sub-domain, have a better or worse chance of getting indexed, getting ranked, and making an impact on the web than a page that gets hosted on a static IP, without any other domain names there, and with a private registration? Does any of that matter? Is it ALL based upon content and link structures?