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Re-installing a wall-to-wall closet
(sh.itjust.works)
The ikea PAX is the easy way out.
PAX without doors can also be combined with sliding doors from some other company in front of them.
It's a free-standing waste of space when the nook for the cupboard is already there?
You're not wrong
If you could find sliding doors of the correct height, you could install 2-3 of them on tracks at the top of the space and skip framing altogether. Another option, if you're feeling a little less crafty could just be using those zig zag room dividers to separate the space and put up closet rods. At any rate I don't think framing the entire wall is completely necessary since all you're really looking for is more of a facade than anything.
Home Improvement