The ruins depicted below were really once a hospital (actually what's left is the chapel and its undercroft). Shortly after the Norman Invasion, the site was first dedicated to the sick and was called St. Peter's, as it was affiliated with the Minster (of the same name). After 1137, a new building (the first was destroyed by fire) was named St. Leonard's and helped the sick until the mid 16th century when Henry VIII dissolved the monasteries.
Bits and pieces of St. Leonard's still stand; in some cases, such as the Theatre Royal, newer buildings have incorporated parts of the ruins into their construction. After Henry VIII dissolved the monasteries, St. Leonard's was used for a while as a mint.
During its heyday, the hospital seemed to be a separate entity from its religious neighbors: St. Mary's Abbey and the Minster. It had four acres abutting the city walls and could house 206 patients and about 20 staff who took religious vows..
Below: what's left of the chapel (about the only remaining room left in the hospital besides the undercroft that supports it).
Above and below are photos of the undercroft (the Brits like this name for the cellar!) which somehow picked up a few sarcophagi.