Constitution states that Vice President has to meet
the eligibility requirements to become President. Bill Clinton is ineligible, having served two terms already. HOWEVER, the eligibility clause was written before term limits were established. When written, that requirement referred only to being a natural born citizen, at least 35 years old, 14 years a resident of the United States, who's never been impeached or participated in a rebellion against the US, all of which Bill Clinton meets.
Of course, it would be easy to argue that the advent of term limits expands that list. However, that question has never been officially ruled on. In the very unlikely event that a two-term president tried to run for the Presidency, it would presumably come up in the courts and be ruled upon. Until then, we can only argue.
My tentative answer is that “eligible” roughly means “elected.” I realize that this is far from perfect evidence — it’s 40 years later than the usage — but the earliest law dictionary that I could find that contained the term, Bouvier’s (1843), defines “eligibility” as “capacity to be elected.” (I take it that, by extension, for appointed offices it would mean “capacity to be appointed.”) If that’s how the term was understood in 1804, then Clinton would not be eligible to the office of president, and thus under the 12th Amendment not eligible to the office of vice president.