There is no reason you cannot have conventional double doors on a closet.
Bracing the Inside Corner
To solve the free corner issue, you can put a small stop, attached to the floor at the center point between the two doors. It needs to be wide enought to stop both doors (probably anything over 1 inch would do) and tall enough to catch the doors at the bottom (3/4 or 1 inch should do, and you want it as short as practical).
The simplist approach is to use ordinary domed door stops screwed into the floor inside the closet. One could probably be placed to catch both doors. These may be prone to a toe stub, but are probably the least likely to cause harm.
If you want to make your own stop, you could use a metal L channel or a wooden block screwed through the top into the floor. Metal is smaller, but may be more prone to hurt if you stub your toe against it. A small wooden block with sloped shoulders might be better. It only needs to be thick enough to hold up (maybe 3/4 inches). All exposed edges sould be slightly rounded with sandpaper to soften the edges for safety.
Latching
The simplist approach to holding the doors closed is to use magnetic latches. You can put them at the top only or at the top and bottom. If you do put them on the bottom, put the latch on the door and the strike plate on the floor stopper (another reason to use a wooden stopper).
As an alternative, you could use a cabinet latch at the top.
Finally, you could use a button type latch set into the tops of each door and the door frame. These are somewhat harder to install, but look more finished.
Handles
In each of these cases, you would use dummy handles or pulls to open the doors.
I have an antique chifforobe that is only 16" deep. I purchased 'petite' or kids hangers from Amazon. they are 14" wide and big enough to hold most adult clothing. Bonus, they are very thin and flocked so that your clothes won't slip off.
Best Answer
Instead of changing the size of the opening, use a pair of 24" doors in the "bypass door" configuration. They'll overlap 4" when closed and you'll have a 20" opening when they're open.
If you want to increase the 20" opening, you can cut the doors to make them narrower and this will still be less work than changing the size of the doorway. Even garbage MDF doors can be narrowed almost an inch on each side, especially if you're not attaching hinges or real doorknobs.