AppleTV doesn't connect with the iTunes store

If you try to sign in, but it reverts back to try the same thing again,

then the issue may be two-factor authentication.

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.

The device is about 13 years old, have you considered updating to an Apple TV HD or 4K model?