I've been a fan of the Sherlock Homes series for most of my life and the stories continually reference a fictional London club called the 'Diogenes Club'. London clubs are also mentioned extensively in 'The Long Firm', which is set in the 1960's. I would be interested to know the answers to the following questions.
What exactly were these clubs ?
How did one join ?
What were the costs of membership ?
Do many of these clubs still exist ?
Were there clubs for all social classes at this point in history or just the wealthy ?
Was this culture unique to London or did other major European cities have gentleman's clubs ?
edit: in reference to class there are also the working men's clubs: http://en.wikipedia.org/wiki/Working_men%27s_club
I highly recommend: Milne-Smith, Amy. London Clubland: A Cultural History of Gender and Class in Late Victorian Britain. New York: Palgrave, 2011. it addresses many of these questions as well as how clubs contributed to gender norms of "the gentleman."
I can answer question 4)
There are still numerous clubs still active, including the Savile Club: http://www.savileclub.co.uk/ and the Caledonian club: http://www.caledonianclub.com/.
There is a list of London gentlemens' clubs on Wikipedia that you might find interesting: http://en.wikipedia.org/wiki/List_of_London's_gentlemen's_clubs
EDIT: CptBuck has covered everything in depth anyway.