One of the requirements of the Veep is to take over as president. Obama is ineligible to be president having already been president twice.
Therefore, he is not eligible to be Veep.
For clarity, Obama is ineligible to be elected VP because of the last line of the 12th Amendment which states:
“But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.”
And it is the 22nd Amendment that makes him constitutionally ineligible to be elected to the Presidency.
It should be noted that the 22nd places a prohibition on “being elected” President, not on serving as President. The House could elect him as Speaker of the House and have him become President if the office of VP is arranged to be vacant when Biden resigns as President.