I want a web directory as a service

Understanding the breadth and depth of the Internet is challenging. Sure, search engines provide an index that allows users to find interesting content, but where is there can I find a breakdown of all websites within a specific category? Google provides search results based on an index, but what if I want to view a (user friendly) index directly? Put another way, if the (U.S.) Library of Congress were to view the Internet as an Information Utility, and was tasked with create a live directory of domains, what would it look like and how could it serve the public?