Closet doors are probably not going to work. Access with the slope will be odd. Curtains are your best way to go. Screens can get awkward to move every time you do laundry. It looks like the lino is also in the entry/ step down. I would try and continue the wood? flooring through out, including inside laundry area. For a look that wont catch the eye, the laundry area should be painted the same as the room. A soft white would work as a second option for me. The goal is to be as seamless as possible, and not catch the eye on any of these details. Good luck!

the top of the laundry area has a inlet and an already installed portion of what I think they used as the sliding screen door at one point.

Paint the walls a nice yellow green and then paint the inside of the alcove where the appliances go a soft turquoise. Don't install curtains. Put in white shelves and baskets and keep laundry supplies in the baskets. Make sure to install the shelves high enough to clear the lid to the washer if it is a top loader. I would do white appliances, not stainless. Love the shelf unit, repeat the colors of the walls and alcove. Paint the shelves themselves white. Get a great rug for the floor to repeat the colors. Don't necessarily have to vinyl the floor just use rugs that can be cleaned.

If you have a front load washer and dryer, you could put a shelf on top all the way across and then add more shelves above w/ baskets/organizing bins overhead like a PP mentioned. (I don't think you have room to have the drawers under the washer/dryer as shown in the pic.)

I also found this pic where they put cabinet doors over washer and dryer and left the top open.