In Ancient Greece, Athenaeum was the name given to buildings dedicated to the goddess Athena; in particular, it was the name of a temple in Athens where poets, philosophers and orators gathered to discuss their work.
During the Victorian era, the term was used for various kinds of academy and learned societies.
There is a gentlemen's club on Pall Mall in London called the Athenaeum, but this is unlikely to be the venue referred to. More probably, it is a reference to the Athenaeum Club which was opened in the Vale of Health in north Hampstead in 1877. This was frequented by political radicals and foreigners, and its upper hall was leased to the Salvation Army in 1882.
It also appears that there was a ladies-only club known as the Ladies Athenaeum in existence at one time in London (possibly between 1913 and 1925).