The qualifications to be vice president are the exact same to be president.
Barack Obama is therefore ineligible to be VP.
Yes, nobody who has been elected President twice can be elected VP. Back in 1964, William F. Buckley, Jr., had the crazy suggestion that Barry Goldwater should pick Eisenhower as his VP.
OTOH, since wee barry has already ILLEGALLY served TWO terms as president, he can NOT be anyone's VEEP, because in case of death/whatever, the VEEP becomes president AND that he and every other two term president may NOT do!