The server still needs to exist

Not necessarily. Consider Freenet:

Unlike file sharing systems, there is no need for the uploader to remain on the network after uploading a file or group of files. Instead, during the upload process, the files are broken into chunks and stored on a variety of other computers on the network. When downloading, those chunks are found and reassembled. Every node on the Freenet network contributes storage space to hold files and bandwidth that it uses to route requests from its peers.