So we have the Hanged Man for gathering around a table, drinking, RPing and looking for work. Perhaps the city now needs a second option for when no work is to be found imminent. How about a location that has various arena-type rooms. For example, it would have sandy areas for sparring, perhaps stone and magic protected rooms for spell duels, obstacle courses for physical competitions and rooms for battling intellects against one another at games of chess. It could also have workout areas and the such. It could be a type of Adventurer utopia where you can practice your skills so you don't get rusty between adventures.
Sessadore! You're forgetting one of the important aspects of 4e: Skill challenges. An obstacle course would consist of say a sprint (endurance), followed by swinging across a ravine (acrobatics) followed by climbing a wall (athletics) and maybe a swim (endurance) and other types of acrobatics and athletics checks. It wouldn't be :21 v. 15, you win, but rather who gets to the success requirement first. Say, 6 successes and you win, 3 and you lose. Whoever gets to 6 first wins or if someone gets 3 losses, they lose. This same pattern could be used to play "chess" or some such.
Skills for games: perception to detect the opponent's moves (or insight rather), history of past wars [actual knowledge of warfare helps win the game?], Endurance checks [who can stand the ennui of sitting at a table forever], and uhm...idk.. I'm sure someone else can think of better examples.
EDIT: And I'm sure the rules of sparring could be decided by the two who are taking part in it. Perhaps a third character could be a judge. I know Les wouldn't want to go toe to toe with anyone. She doesn't believe in fair fights because they don't exist in reality. Also, warlord's aren't all that effective alone..