That fellow’s blocking Civil Service game (5)
I believe the answer is:
chess
'game' is the definition.
(board game)
'that fellow's blocking civil service' is the wordplay.
'that fellow's' becomes 'hes' ('he' refers to a man).
'blocking' indicates putting letters inside (to block can mean to plug a gap in something).
'civil service' becomes 'CS' (abbreviation).
'hes' inserted into 'cs' is 'CHESS'.
(Other definitions for chess that I've seen before include "that has over thirty men on board" , "Ancient war game whose name may have derived for the Persian for 'king'" , "activity aimed at downfall of a king" , "This is played on board" , "Game with kings and queens" .)