Is there a way to DVR shows either through basic cable (encrypted) or OTA? The integrated guide is nice and will allow to select "View", but there is no option to record. Is there a way to enable that option by plugging in a hard-drive or setting up an app?
Coming soon, apparently, e.g. on UK sets:
Best to check the update info for your model & region.
https://www.kickstarter.com/projects/1275320038/hdhomerun-dvr-the-dvr-re-imagined/
you need a compatible storage device/host for siliconDust tuners and your ATV unit will work as the playback/control endpoint. don't think they have an option yet for the storage to be hosted by the ATV itself.
your built in tuners wouldn't work, though. that'd be on sony to implement a similar stack to sd's. for those that made their android tv ala-cart, with a separate box, some SD tuners and a monitor - they'd only need the right storage piece to gain dvr support.
it's early days for that project though, so who knows how it will work out (but they do seem like the most likely option to me though)
The answer to your question is incredibly complicated. Is it possible? Maybe... Is it as easy as downloading an app? No..
You need a tuner of some sort, connected to a dvr system of some sort(pc, Nas etc) then you need a playback device(android tv).
However, if your cable is encrypted then you can only use a Windows machine for the dvr backend with wmc as the front end. Eliminating your AndroidTV.