The World Wide Web never "runs out of space" because webmasters (people who have websites) are generally willing to pay for a website. And that means that people in the web "hosting" business have the cash to pay for more web servers if they need them.
In practice, web servers usually aren't exactly like your computer. For business reasons, they are usually built to fit into racks in "data centers" that have very fast connections to the rest of the Internet. This way, web "hosting" companies can achieve economies of scale by hosting many websites in one facility.
But there is no fundamental difference between your computer and a server in a rack at a web hosting facility. In fact, it is possible to run a website on your own computer at home, although that is usually not practical for serious websites.
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