You need an ISP to connect to the internet. Verizon just supplies a DSL
line to a MUX that goes nowhere unless it is routed. Either to them or a
third party ISP. You don't have to use the ISP's services though as long
as it is cool with them. It is not cool with Comcast or Verizon.
Speakeasy, DCAnet, DirectDSL, etc. don't care much what you do. You can do
your own domain, mail, web etc. serving. They just supply the pipe to the
internet and and IP address or addresses to do it. Of course if you don't
want to do all that they supply services also. Most also supply a free
dialup in the event the DSL were to go down.