This is across the court from the carpet museum, a “hazine” or strongroom of the Hagia Sophia. On the web one can find more when looking for "Skeuophylakion" i.e [it] was a place where the sacred objects of the religious ceremonies were kept in the early Byzantine times. It is also thought that the building was used by the clergy before they entered the church for the rites. Later, it was used as a store of the alms house in Ottoman Period. The Skeuophylakion dates back to Justinian Period.