Jack Balkin uses the question as a springboard into an entertaining discussion about how text and structure illuminate the issues; turns out, under Article II, sec. 1, clause 4, no one alive today is eligible to be president of the United States!
I come to something like the opposite view.
In pertinent part, Article II states:
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.
As I pointed out in a comment in the thread to Balkin's blog, this means that one is eligible to the office of President if one is either (1) a natural born Citizen or (2) a Citizen of the United States. But then everyone is a Citizen somewhere. Therefore, everyone is eligible to the office of President (as long as they are willing to adopt the Constitution, of course).
SIDE NOTE: The restrictions the constitution sets on eligibility to the office of president are all limited in scope to "persons." Therefore, it would also appear that nonpersons are eligible to that office without restriction.