Peeled in a jar with pickle juice.
With the shell, covered.
Eggs will not only pick up flavors from your fridge, but will also give off flavors to your other food once they're hard boiled. The shells are porous so they will pick up flavors no matter what, but shelling them will make it happen faster. Plus, they'll dry out faster.
They get pretty gross pretty fast sans shell, so I always keep them in the shell until it's time to eat 'em.
Without shell in closed tupperware container.
I boil 2 dozen at a time, and it is more convenient to go ahead and remove the shells all at once (and my wife is more inclined to eat them too if they are already shelled for her). If you crack the shell then let eggs set in an ice-water bath for a 10 minutes or so right after boiling, the shells come off pretty easy.
I will eat them up to a week later, but they don't taste as good at that point and the yolks are kind of dried out. Usually they get eaten in 4-5 days though and seem mostly fine at that point.
They last longer IN the shell, but we go through them so quickly that I can boil up a dozen on Sunday and they'll definitely be gone before there's any chance they'll spoil -- so I peel and store in a covered container for the convenience of it.