The answer to your question as stated - could Obama be appointed to the Cabinet? - is a resounding yes for the reason that TDAS04 discusses in their post.
The answer to the broader question underlying your post - could Obama be permitted to ascend to the Presidency once again? - is a bit more tricky & would likely become a question for the Supreme Court if ever necessary, because a very strict reading of the Constitution would actually allow it: the 22nd Amendment only says that a person can't be elected President more than twice, without saying anything about whether or not a person can serve more than twice. That is, if the Court were to take into account the words exactly as they're written, then the answer to your question would be yes. If they were to take the clear intent of the 22nd Amendment into account, then the answer would be probably not.
But until that happens, who knows?
3 USC 19(e) could preclude this. 3 USC 19 provides the succession in case of a double vacancy.
3 USC 19(e) excludes those not eligible, including those too young, not a natural-born citizen, and presumably those not eligible to be elected.
