Hopeless sort of case one finally checks out? (6)
I believe the answer is:
basket
This clue is a double definition.
'hopeless sort of case' is the first definition.
(as in a basket case)
'one finally checks out?' is the second definition.
(a basket is taken to a check out in a shop)
(Other definitions for basket that I've seen before include "Container of woven twigs etc" , "Container made of interwoven cane or wire" , "Container with handles" , "Hamper" , "Wicker carrier" .)