Again we bump into some problems of literalness, though, because some would argue that a person who has already run up against the limits of the 22nd Amendment isn’t “constitutionally ineligible” to be President (i.e., that person doesn’t fail to meet one or more of the requirements specified in Article II of the constitution, such as being at least 35 years old or a natural-born citizen of the United States) but is merely constitutionally ineligible to be elected President. Under this interpretation, a former two-term President would still be eligible, under the law, to obtain and hold the office of Vice President through election or appointment to that position.
Ultimately, the only answer to this hypothetical is that there is no answer: this is an interpretive legal issue that, should it ever arise, would have to be decided by the judicial branch of the U.S. government.