Shakespeare's A Midsummer Night's Dream, as the title suggests, is set in a dream word, albeit one with classical and mythological influences. Because the majority of the action is not located in an historical setting, but an imaginary one, the set and costume designers for the play can visualize it creatively.
For costume design, as the fairy parts are often highly choreographed, and even danced, costumes should allow considerable freedom of movement. Personally, I would adapt classical Greek clothing, albeit with shorter tunics of the sort used in ballets with ancient Greek themes, in white, with perhaps colored edges, as white allows most scope for interesting lighting effects.
For staging, as the Royal Shakespeare Company has done, I'd use abstract multi-level settings, perhaps just platforms and moveable blocks, and rely on lighting and imagination to create physical setting. This type of abstract set gives the choreographer the most freedom to use spatial relationships effectively