Sleep on family's sheets (7)
I believe the answer is:
napkins
'sheets' is the definition.
(napkins are sheets of material)
'sleep on family's' is the wordplay.
'sleep' becomes 'nap' (a nap is a short sleep).
'on' says to put letters next to each other (in a down clue, letters appear on others).
'family's' becomes 'kin's' ('kin' can be a synonym of 'family').
'nap'+'kins'='NAPKINS'
(Other definitions for napkins that I've seen before include "Pieces of table linen" , "Serviettes" , "Cloth squares to protect clothes while eating" .)