i reset my appletv to fix another issue, which also fixed this, sometime in february. i'm now seeing it a few times a day. i tested the same stream via plex app for a whole day and never had an issue. my appletv reports more than enough space via the tvinfo app. seems i have to use that as ios doesn't give a good breakdown, but i'm not sure how accurate it is. i have plex/channels and aerials installed. i tried clearing them out and it still happens. sometimes it happens after only 30 mins-hour of live playback. and yes i believe it also happens if you pause playback for an extended time.
it's not the best testing method using the plex comparison, but it doesn't throw the device disk full error from what i can tell. i'm not sure it's a channels specific issue, an ios per-app-cache issue, who knows. searching around though, i see other complaints of this issue on the apple forums and it's not people using channels.
the only fix i can find is the reset, but it sucks resetting expensive hardware every few months.