Serve Jellyfin from a 1TB WD Element SSD?

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No, you need a server, storage, and client. These all need to be able to communicate over a network. In the scenario you've described you're missing the server and networking. You could run Jellyfin server without the web client using a recommended SBC for the smallest footprint and low power draw or step it up to a mini-PC if you have other use cases. You'd also need a cheap routing device to create a local network of some sort.