There is a chess club called Jaque Mate on Santiago del Estero between Salta and Estados Unidos. The other one is on Paraguay but I don't recall the cross street (recoleta area I think)
Paraguay 1858. That's the real chess club of Buenos Aires and is over a century old. It has three storeys. And inside you will find furniture and memorabilia related to the Alekhine-Capablanca world championship match of 1927 and the Fischer-Petrosian world semi-final match of 1971. Plus some stuff from the San Luis Tournament of 2005. Most evenings some players can be found for 5-minute or 10-minute games. For non-club-members, there's a charge of 7 pesos for the evening.
There's often a chess bookstall downstairs. And when big tournaments are being held, there are two.
Plus food and non-alcoholic drink are for sale.
If you're not a club player and are not au courant
with the latest wrinkles in the Sicilian Sveshnikov or Winawer French, you can still get a decent game in the parks on weekends (I'm still smarting from getting beaten twice in such park encounters).
The website: http://www.cargentinodeajedrez.com.ar/