Quiet spot, one in outskirts of Aachen, for a museum (9)

'museum' is the definition.
(the Ashmolean Museum in Oxford)

'quiet spot one in outskirts of aachen' is the wordplay.
'quiet' becomes 'sh' (both can be said to tell someone to be quiet).
'spot' becomes 'mole' (spot or blemish on the skin).
'one' becomes 'a' (a thing is one thing).
'in' is an insertion indicator.
'outskirts of' says to hollow out the word (remove centre letters) (only letters on the outskirts).
'aachen' with its middle taken out is 'an'.
'sh'+'mole'+'a'='shmolea'
'shmolea' inserted into 'an' is 'ASHMOLEAN'.

'for a' is the link.