There are requirements for elected, national office, and those requirements are most restrictive for the Presidency. To my knowledge, no questionable individual has ever won a majority of electoral votes, prior to Obama. Questions can arise, and did in fact arise, because the factuality of certain claims remained unclear, and still remain unclear.
There were questions all along. There were Hillary supporters questioning Obama's eligibility prior to his being nominated as the Democrat candidate. There were attempts at legal actions directed at both McCain and Obama. The whole thing didn't just fall out of the sky in June of 2008, as several have spun on this thread.
Regardless of the philosophical exercise, the factual basis of Barack Obama's eligibility has been in question for some time, certainly well before the electoral vote was certified. I recall widespread mystification on FR, as to why no one had standing. It was explained away that the matter was not "ripe," or some other rationale, that the matter had to wait until this or that milestone had been passed. Those milestones included party nomination, the popular vote, the electoral vote, the certification of the vote, the Oath Of Office ... always something later.
Now, the spin is that it's too late to be actionable through any means other than through Congress, impeachment I assume.
The plain words of the Constitution state that the Vice President becomes acting President, if the President is ineligible.
How that is to be accomplished, or even how such determination is to be made, is not a matter that is defined. I've contended that determining Constitutional disability and then following the line of succession is the route to take. I see no means of reaching such a determination, outside of SCOTUS.
Barack Obamas' birth status is in doubt. Such doubt is not a recent phenomenon, as regards Barack Obama. Unresolved doubt precludes the requisite status required for office.