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Jellyfin server isn't available for Android. If you installed Linux on your Shield, then yeah it would run, but none of the Android streaming apps would, so meh.
Like someone else said, just get a used office computer for cheap and run your server off of that.
The hardware transcoding from even somewhat older Intel iGPUs is rock solid.
I have no idea what Nvidia shield is like tbh... it might violate their ToS.
I don't think it's practical. Sure, you could install termux and use that to run Jellyfin, but you won't have a pleasant time.
The better play is, to play spend 30 Bucks on a refurbished thin client and use that as your server. If you install Open Media Vault on there you'll never even have to touch the command line. (You'll have to install OMV-Extras on top of Open Media Vault for that).
do you have any suggestions for these refurbished thin clients?
I'm using a Dell t520. That works really well for me.
But honestly: pretty much every client in that price range should work. Just check whether the (embedded) graphics chip can hardware transcode your media beforehand. And you probably won't run the next Netflix on there but it's more than enough for a family usecase with multiple streams simultaneously.