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No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President;

So how can someone not eligible to the office "hold" it?

Because the Twelfth Amendment clearly states who actually IS the President:

The President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates and the votes shall then be counted;- The person having the greatest number of votes for President, shall be the President...

Obama received the greatest number of votes for President. He IS the President, and therefore "holds" that office until such time as Congress shall remove him (barring expiration of his term, etc.).

By that argument, someone could walk in off the street and "hold" the office

Not without having received the greatest number of votes for President they couldn't.

The person having the greatest number of votes for President, shall be the President...

But the person in question must have first met the Article II sec. 1 clause 5 conditions for eligibility for the office.

"No Person except a natural born Citizen, or a Citizen of the United States, at the time of the Adoption of this Constitution, shall be eligible to the Office of President; neither shall any Person be eligible to that Office who shall not have attained to the Age of thirty five Years, and been fourteen Years a Resident within the United States."

Not without having received the greatest number of votes for President they couldn't.

True, but also not without being a Natural Born Citizen. The Constitution sets out many requirements. All must be met.