Summary: Is it legally possible for Barrack Obama to run as Joe Biden’s Vice President?
Let’s start with what the eligibility requirements are to be Vice President. the 12th Amendment to the US Constitution  states:
no person constitutionally ineligible to the office of President shall be eligible to that of Vice President of the United States.
Article 2, Section 1 of the US Constitution  states the following eligibility criteria for being President:
No person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any person be eligible to that Office who shall not have attained to the Age of thirty-five Years, and been fourteen Years a Resident within the United States.
To be Vice President you have the same requirements as to be President. The 22nd Amendment  adds another requirement to be President, which is the term limits for office, you can’t be president for more than two terms. What this means is that any person who has been President for two terms is legally ineligible to be President, and since you must be eligible to be President to be Vice President, Barrack Obama isn’t legally eligible to be Vice President.