Most people know that there is a two-term limit on the office of President and therefore believe that Bill Clinton is ineligible to become Vice President because he is ineligible to be President. It appears that view is incorrect. The Constitution and applicable amendments forbid a person from being elected to the office of President more than two times. At least on it's face, the Constitution does not forbid a person from becoming President by any other process, e.g., presidential succession. As such, while Bill is ineligible to be elected President, he is not ineligible to actually be President. Bill is therefore is eligible to become VP and can indeed be his wife's running mate and assume the Presidency if Hillary is elected and thereafter cannot fulfill her duties. Scholars generally agree on this analysis, but the issue has never been formally addressed by the Supreme Court.
There are a couple of other relevant factors, as well: (1) Bill would only be eligible to be President for a maximum of two years (there is an effective 10-year limit on how long any individual can be President) and (2) Bill and Hillary would have to declare that they are inhabitants of different states as the President and VP cannot be from the same state (granted, this may not be a problem given the apparent "quality" of their marriage!).