LAST LINE OF THE TEXT OF 12th AMENDMENT
"But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States."
Obama is not eligible to be president as he has served two terms (22nd amendment). Therefore, he is not eligible to be vice president.
Allowed and eligible seem to have little consequence these days.
Reminds me of The Princess Bride, "Inconceivable!", "I do not think this word means what you think it means..."