I think the correct and nicest way to complete your room is to have your handyman build a wall to frame and install real doors (that come along with the door frames).
In the master bedroom, you will need to install two sets of double doors. In other words, it will appear as though there are two closets when the doors are closed but it’s one wide closet on the inside.

Closet doors keep the dust off the clothes. So build a frame for real doors or sliders.

You could add a track to the ceiling and put slidding doors in.

Lema · More Info

@Shifra Lauber I don't think two double doors would work because then I couldn't access the shelves in the center properly.

You could install triple sliding panels or double swing out doors in the master BR, and double swing out doors in the other BR.

When you are a contractor, you see all kinds. We had a customer who insisted that all the closets throughout the house were open because "they are do pretty to look at." They claimed that they often woke up a night just to look at the clothes hanging in the closet. I am still laughing about these customers 5 years later. If I walk up at night, the last thing I am doing in my bedroom is look at my clothes. LOL. Can your handyman handle framing, drywall, painting, electrical and carpentry? Most reach in or bi pass closet doors are 6'8" and you probably have 8-9' ceilings. You also have several cans above the closet area. I would remove the cans except one light which should be switched outside the closet. Bi pass would be easier because they function off a track but you will not have access like a full reach in.

Agree with all the commenters - for the master, frame out a real closet and add real doors with a style of your choosing. However, for the baby room, I wonder if you might like a cute curtain instead? It would be a nice way to bring in another soft element - would save money in the short term and you could always add real doors later if you decide they are needed.

No framing / triple doors hanging from ceiling mount rail as shown by @modpod. Previous home had triple doors with tall chest of drawers within closet = full access side-to-side & top shelf, no problems. A very clean & modern look.

Ceiling mounteed sliders