So is someone who has served two terms.
But the XXII Amendment does not make anyone ineligible to anything.
But the 12th Amendment does: "But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States." Obama is constitutionally ineligible to be elected president, he cannot constitutionally be eligible to be elected vice-president.
That is correct.
Are there ways to become President without being elected?