After a US President serves two four-year terms, can they run again after four to eight years being out of office?

No, a US President who has served two four-year terms cannot run for the office of President again, regardless of the time they have been out of office. The Twenty-second Amendment to the United States Constitution explicitly states that "No person shall be elected to the office of the President more than twice"45.

This limitation applies even if the terms are not consecutive, as seen in the case of Donald Trump, who was elected to two non-consecutive terms and is barred from seeking a third term35. The amendment does not provide for any exceptions based on the amount of time a person has been out of office.