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The Market Hall was constructed in the latter part of the 17th century, as a way to provide shelter and protection from the weather for salesmen and stallholders in the town's regular market – the precursor, Booth Hall, being declared insufficient. The building was constructed with a large, open-plan ground floor with wide open arches, to allow easy access to the stalls. The first floor also housed a number of rooms, which after 1694 served as meeting rooms for any organisations which wished to rent them.[1]

From the early 18th century to 1848, one small room in the Hall served as a "lock-up" – a small room which was used to hold prisoners before magistrate trials. From 1833 until its closure in 1848 the room was declared a "disgrace" – in 1842, it was reported that the room was "8ft 8ins by 3ft 8ins" and at one point housed thirteen people at once.
The building was later Scheduled as an Ancient Monument in 1936 to prevent its demolition, and in 1938 work was set about to repair and restore the building.
The Museum was classified as a Grade II* Listed Building in 1953, after a short spell as a Civil Defence Store during the Second World War (Wikipedia)