When you go down the elevator at the Greenwich Hotel, it feels like the doors open into another realm. The dim lobby of the The Shibui Spa is lit by Japanese lanterns, and the tone of the staff is professionally, soothingly hushed. To begin, I was escorted into the locker room, where I was offered sake and a sparkling yuzu drink. After a dip in the pool—which was warm, but not too warm—and a steam in the locker room, I followed my massage therapist through a winding otherworldly maze to the treatment rooms.