Bathrooms need constant ventilation, and therefore I would put a louvered door on it. They come pre-hung in their frame. That rough opening needs to be between 1/2-1” wider than the frame/jamb set, depending on your ability to frame a plumb wall.
The framing itself is conventional framing and since you’re not likely to hang clothes on hangers inside this closet, the inside “clear” dimensions don’t have to be 24” deep.
It’s easier to direct you to framing videos on YT than to teach you how to frame in words
I would use a water-resistant drywall if this bathroom has no exterior window. I would also advise buying a new exhaust fan, if it is less than 80 CFM. It should vent outside, not into the ceiling joists.
Use a pantry cabinet, make it built in by adding base board and maybe crown. Very easy and takes up a lot of less space. Can.finidh it like your cabinet or walls.
Did you get my previous answer? I don’t seee it here. LMK. Thanks
Hi Carole, you can purchase or build corner shelves and hang a tension rod to "hide" it behind a curtain