And it's misleading to talk about the entire web. Google may index billions upon billions of pages, but only a small percent actually get shown to the user, or stored in memory. And I would say that a smart kid with a few computers or a couple of bucks can build the index.
Then the hard part starts. Machine learning to correct search placements, based on features like clicks and time in index. Algorithms to detect spam. Anticipation of user data needs. And all the nitty gritty of linguistic analysis to better sort your index.
These types of things require new and innovative thoughts. In my mind, building a search index is just standing on the shoulders of the giants that came before. I'm not saying Google isn't amazing or that building a search index isn't a great homework problem, but the basic part of getting, storing, and serving the data has been engineered and can be engineered again at will.