Story about horse carrying European ancestry (7)
I believe the answer is:
lineage
'ancestry' is the definition.
('lineage' can be a synonym of 'ancestry')
'story about horse carrying european' is the wordplay.
'story' becomes 'lie' (I've seen this before).
'about' is an insertion indicator.
'horse' becomes 'nag' (nag is a kind of horse**).
'carrying' is an insertion indicator.
'european' becomes 'e' (abbreviation e.g. EU).
'lie' placed around 'nag' is 'linage'.
'linage' enclosing 'e' is 'LINEAGE'.
(Other definitions for lineage that I've seen before include "Ancestral line as seen in eagle" , "Stock" , "Old family members" , "Direct descent" , "The strain" .)