the full phrase:
"and if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President; a quorum for the purpose shall consist of two- thirds of the whole number of Senators, and a majority of the whole number shall be necessary to a choice. But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States".
It is in a case of the senate electing the VP, not general elections
But the context of the *entire* article deals with the whole electoral process, both in the EC and and Congress, of choosing the President and VP. Therefore, that sentence could be (and probably would be) construed ("construed" is such a cool legal word) as applying to the general eligibility for VP.
