We stayed at a modern Jolly Hotel in Siena and our room overlooked the soccer field. It was very nice but not in the historic center, just nearby within walking distance.
Visiting Siena another time, we stayed in a convent built into a hillside in the old part of town. This was in the historic center and I think near the church that houses the relic of the patron saint of Siena.
I can't think of a hotel overlooking the Campo but we enjoyed the convent location, the view down the hill was lovely and the rate was very reasonable.