Software: 5.2 (6010.96)
A newer version system software should be available for that model (v6.2.1 build 11D258 from 2014-09-17). The update file is still hosted on Apple’s servers (verified). To update the software on Apple TV (2nd or 3rd generation), go to Settings﹥General﹥Update Software﹥. If there's an update, download and install it. Don't disconnect or unplug your Apple TV until the update completes.
"Account Sign-in Failed. The iTunes Store is currently unavailable. Try again later"
Can I only be logged in to one at a time?
Normally, you can be logged in to as many Apple TV boxes as you have. Not sure if Apple changed something recently regarding legacy devices; I can’t check that. We don’t see many users with this model anymore.
It is advised to reset any second hand device first, to purge any customization and configuration settings from a previous user. Reset or restore your Apple TV to its factory settings - Apple Support. Then set up “as new”.
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If you have an Apple TV (2nd or 3rd generation) and two-factor authentication for your Apple ID, then you may be asked to type your password followed by the six-digit verification code directly into the password field. Additionally, there is a time limit, after which the code becomes invalid.
Try this: First enter the characters of your password (do not submit yet), then manually request a new verification code, then append that code to the password (no spaces or other separating characters), and then submit. That should help you get passed the time limit.