But Obama and Clinton are NOT ineligible to BE President, they are ineligible to BE ELECTED to the Presidency.
The 22nd Amendment is a restriction on the 538 Electors for President of the United States, nothing more.
Obama could “run”, he could appear on the ballots of all states that list candidates by name, and millons of Americans could vote for Electors pledged to him.
The 22nd would simply void any electoral votes naming Obama as President.
It doesn’t say anything else. It would have been easy (and smart) to draft the 22nd to say, “no person having served...shall be, after his service, eligible to the office...”, which would bar the VP slot as well, but that’s not what the amendment says.
so then, it comes back to whether the Clintons or Obama, or anyone else, would try to work around these restrictions. It’s hard to put anything past the Obama/Clinton political cabals.