Having had some time to research them, the museum’s senior curator of archaeology Francis Grew knows a little more about the vaults’ history and purpose. They were built along with the general market in the 1880s, partly to help hold up the busy Farringdon Road above, but also, he thinks, as storage space for the huge volumes of goods that were coming in and out of the markets on what was then the busiest stretch of railway in Britain.
Very plausible.
I wouldn’t be surprised if a part of it was used as in ice cellar.