I would call it a balcony. A closet seems to be a better use of space, although it does bring natural light into the second floor hall. If you are good with closet space, I'd leave it.

Call it a nook, then make it a cozy spot to read.

LOL jwp. Would that make it the doghouse?

haha

Ha, yes!

Very weird use of space. I'd close it off and make it a closet.

In a more refined time it would be the perfect place for a writing desk and chair.

In all honesty, I think it can be whatever you need it to be. It looks like a sitting area to me. Possibilities are endless and could be directed by your house layout and needs.

Put a big comfy chair, small table, lamp, and read a book or drink coffee/nightcap there.

Maybe use it as a dressing area or makeup area with those great windows.

Recliner and small tv and mini fridge to watch tv after hours.

Game table for your nightly battle of scrabble.

Lots of options!

Yes, a little office space is good too!

I kind of like it. I have things in my bedroom that sort of need to be in there so my kids don't get in them, but I'd also like them to not be in my bedroom. That space would be a way to have it close but out of sight.

Do you have to go through the master to access it?

Here's kind of a crazy thought. You could hang curtains from the ceiling in front of the railing. When you wanted privacy, close the curtains, otherwise leave them open for the light from the windows.

It's called a Juliet Balcony. Neighbors had one that went from an unfinished attic and overlooked their family room. The next owners had the space walled off and the balcony removed.