Apple TV keeps resetting itself

Thanks for the additional information.

I can see no relation with a specific app. Content from a sports app shouldn’t be much harder to process than a e.g. a movie. No stream should be able to send anything that forces a reset unintentionally.

• Do mind the heat on the devices. Do they have means to dissipate heat? Are they on a hard flat surface with enough free space on all other sides for air flow? It shouldn’t have to be a lot, but you shouldn’t box it in tightly either.
• Try turning it really off for a while (30 sec), including disconnecting all cables.
• If that doesn’t work, try restoring to factory settings from the menu. Settings > General > Reset > Restore
• If that doesn’t work, try restoring through iTunes.
• Confirm the software is up to date (for 3rd gen: Apple TV Software 7.2.2 from December 2016)

Which model are your Apple TV devices? Identify your Apple TV model - Apple Support

Since this happens on both AppleTVs I have, I’m thinking it is the network.

Maybe. The intermittent nature would suggest looking at your network. However, resetting of the device shouldn’t be invoked by anything on the network, though. That could point to software that requires a reset, or hardware that requires a reset, both as a result of an unrecoverable error.

Both of them are connected by 10BaseT Ethernet.

Are you sure? A 10 Mbps home network seems limited today, with 100 Mbps and 1 Gbps network being far more common. Two streams of a lowly 6 Mbps (~720p) wouldn’t be able to go through you network.

The Helpful response, linked above, suggests a few things, but the only one that makes sense for my situation is the DNS settings. But where should I be looking at DNS? Why would everything else on my network be fine, but DNS would hose the AppleTVs?

It is advised to set IP and DNS to automatic, so that the Apple TV gets its network settings from the router, which most likely uses ISP settings. This eases setup and avoids all kinds of user error and unsupported setups.

I can try restoring and restoring through iTunes. What's the difference?

Regular restoring will remove all (custom) information and set the device back to factory settings, but keeps the system software (as that is functional while doing this).

Restoring through iTunes will replace all content with a fresh copy, using a different device (Mac/PC) to control it. Even the system software gets replaced.