How to backup photos saved on Apple TV hard drive?

You can't back up content that is on your Apple TV, however they must have been somewhere else before you added them to the Apple TV, you can back them up from there.

The ATV 4 doesn't "store" them rather streams from iCloud photo share.

I Think you can save from iCloud though but not positive

this may be sone help...

iCloud Photo Sharing FAQ - Apple Support

Can photos and videos be saved or downloaded from a shared album?

Yes. After photos are shared, anyone who can view the shared album can download and keep them permanently. Videos can only be downloaded to a Mac or PC. On an iOS device, videos are streamed from iCloud and can't be downloaded.

• On a compatible iOS device, you can save photos to your Photos tab or Camera Roll.
• In iPhoto or Aperture for Mac, you can add or copy shared photos or videos to your Library.
• On a Windows PC, you can copy photos and videos from a shared album to any folder.
• On the web, you can download photos and videos directly to your computer from the detail view of each photo or video.

The Apple TV stores photos it uses for the screensaver, (it doesn't store other photos from your Mac). Unfortunately, short of pulling the Apple TV apart, there isn't any way to copy them from there. Also I suspect they aren't full resolution photos.