The Internet isn’t a big pool or one centralized server – anyone can host something on the web. You could host a website off your home computer if you wanted, and people could access it from anywhere.
It would be slower than accessing a mainstream website, but assuming you were willing to relocate your house closer to an exchange or to a T1 access point, you could eventually serve more and more people simultaneously.
When your computer can no longer simultaneously serve the number of people viewing your website, you add another computer and a load balancer, which shares the request between the two.
Eventually you might think about using server-grade hardware, which is just computers with really fast hard drives and lots of RAM (depending on exactly what type of website it is, there’s lots of different hardware decisions you could make to make your website run faster).
This is a simplified explanation, but I hope it helps.