Could the internet ever run out of space?

The short answer is no. The internet isn’t a thing in itself, it’s a concept. All the computers in the world that are connected to the internet are themselves part of it. As the individual storage of each web server gets upgraded, so the capacity of the Internet gradually increases. There are always limits imposed by current technology – to the total of unique internet addresses, for example, or the number of simultaneous connections that each server will accept. However, new protocols and hardware are continually developed to  extend these limits, so the internet will inevitably continue to grow for as long as we need it to.