Story involving very rare organ (5)
I believe the answer is:
liver
'organ' is the definition.
(liver is a kind of organ)
'story involving very rare' is the wordplay.
'story' becomes 'lie' (story can mean a fabrication).
'involving' means one lot of letters goes inside another.
'very' becomes 'v' (abbreviation).
'rare' becomes 'r' (listed by Oxford Dictionary of Abbreviations).
'lie' enclosing 'v' is 'live'.
'live'+'r'='LIVER'
(Other definitions for liver that I've seen before include "Part of body" , "Dark reddish brown" , "Viler (anag.)" , "Glandular organ in the abdomen" , "Organ damaged by excess alcohol" .)