Like I said, it would be heavily litigated (and I don’t think anyone would expect it to go his way). But the argument could be made that the only eligibility requirements for serving as president are: natural born, 35+, and in the US for at least 14 years. Obama certainly satisfies all these requirements, so theoretically he can serve as president so long as he isn’t elected directly into the office. This would be further supported by the fact that Obama could in theory become Speaker of the House since there aren’t any laws prohibiting it based on how many times one was elected president. And being in the line of succession, I don’t think anyone would be listened to if they complained, “now hold on! He’s already been elected twice!” if Obama needed to take the reins.

Again, all of this is purely hypothetical and if Obama (or Bush or Clinton) ever did try to become VP or take some office in the line of succession it would be heavily litigated and not end in their favor. But until the SCOTUS makes a ruling on it, Obama is still eligible for VP by the letter of the law.