"Men with trays on their heads went busily from staircase to staircase"

Here, Virginia Woolf is referring to the college servants who were employed to wait on both undergraduate students and academic staff.  In Brideshead Revisited by Evelyn Waugh (which is partly set in 1920's Oxford), both Charles Ryder and Sebastian Flyte have servants known as scouts who are deployed on such diverse activities as clearing up vomit and arranging flowers!