Went out with, despite being antiquated (5)
I believe the answer is:
dated
I believe this is a double definition.
'went out with' is the first definition.
('date' can be a synonym of 'go out with')
'being antiquated' is the second definition.
(I've seen this before)
'despite' is the link.
(Other definitions for dated that I've seen before include "Revealed the age of" , "Went steady" , "Antiquated" , "not without romantic engagements" , "Old fashioned" .)