Eligible means qualified, entitled: “Only people who have passed college are legible to participate in the job fair.”
Ineligible is the antonym of eligible and means not qualified or permitted to participate: “Because of his low marks, he was ineligible for the job.”
Illegible has nothing to do with qualifications and everything to with handwriting, print etc. it means unreadable: “Students write so carelessly that their writing is illegible.”
Legible, as we can guess easily, is its antonym and means writing which is clear and plain enough to be understood: “Answers written by a legible hand fetch greater marks.”
Who would be the most eligible for the new position: The person who had the most legible handwriting but had poor Public Relations skills, or the one who had illegible handwriting but who was great with people?
Finally, it was decided that the person with good handwriting but poor PR skills was ineligible for the position.