Source : Webster's Revised Unabridged Dictionary (1913)

Ineligibility \In*el`i*gi*bil"i*ty\, n. [Cf. F.
   in['e]ligibilit['e].]
   The state or quality of being ineligible.

Source : WordNet®

ineligibility
     n : the quality or state of being ineligible [ant: {eligibility}]