This has also happened to me, it is super annoying when the internet goes out (oh the humanity!) and if you’re not set up ahead of time for the event it’s even more annoying. So I have 2 approaches -
a) have your local plex clients and server setup in the event this happens (it’s way more of a pain to set up without internet!) - lots of resources on setting this up properly out there but here’s a few to get you started:
https://support.plex.tv/articles/200890058-authentication-for-local-network-access/
just google “plex without internet” - there’s videos, etc etc etc
b) have plex alternatives set up - we use the roku media player app for roku clients to browse to our hdhomerun device (for tv’s without an antenna/cable line) and the VLC app for firetv and chromecast (android based) devices. In VLC go to “Browsing” and then “local network” and you should see your hdhomerun there, if you marked any of your channels as favorites (which is what we do), you’ll find them in the favorites folder. There is also a HDHomeRun apps for clients - but they seem to require internet access to use reliably (which is useless when there is no internet!)
Hope this helps!