Not that long. A day, maybe, before they start to feel pretty dead for regular recreational play. As the level of play goes up the tolerance for old balls goes down. In top-level pro matches they get out fresh balls several times in a match.

You can buy storage canisters with pumps or other pressurizing devices that will keep the balls fresh longer. (And may even repressurize dead balls. I've never tried that but it seems possible.) Still even at my very low level of amateur play, in my ladder league it was expected that each player would bring a fresh, unopened can of balls.