I always wanted a space like that to make a sweet pad for my dog, with a fire-hydrant lamp, photos on the wall of his dog friends and comfy bed, small corner fireplace etc...
Judging by the spacing of the bannister rails that space looks to be ~36x30. My dog's bed is 26" across, leaving plenty of room for the other stuff. I mean look for a dollhouse fireplace on line, they're pretty small, and the lamp could just be a rubber fire hydrant toy with the lamp mount jammed in.
When my son was younger he was extremely scared of thunderstorms. The closest under our stairs is fairly large and is in the middle of the house, so with a noise maker he couldn't hear the storms in there. We ended up putting a cot in there and decorating it with a bunch of Harry Potter stuff. Any time there was rain he would just hang out in there until it was over. At one point he actually preferred it to his bedroom.