On my trip to Florence, I had stayed at a convent which was really a nice experience. Are there convents that accept guests in Paris?
You may be interested instead in staying at the Hotel-Dieu Hospital, right beside Notre Dame in Paris. The rooms on the 6th floor used to be for patients' family members, but a number of years ago they began welcoming tourists. I have always wanted to stay there. When you enter the hospital doors, you can see a beautiful courtyard garden, and anyone can go and just sit & relax there on a hot summer day.
www.hotel-hospitel.com