I have a coat closet that long ago had double sliding doors and now has no doors at all. I would like it to have double swinging doors, like french doors but without glass. I found a couple of pictures to prove that it has been done before, but they lack the detail necessary for me to reproduce the work.
Normally a single door has a piece of stop molding around the frame on three sides to keep the door from swinging past the "closed" position. With double doors that lack a center column, there can only be a stop along two sides of the door. This means an accidental kick to the "free" corner of the door will put undesirable bending loads on the door. How is this normally solved? French doors have a central column attached to one of the doors, but that means one door must be opened first and that's undesirable for a small closet.
An ordinary door lockset won't work here because both doors must have a handle. I have seen "dummy" handles that don't have a striker bolt. I think that'd work but if those are used, how does the door stay closed?
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.
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.
In each of these cases, you would use dummy handles or pulls to open the doors.
You can use any of the options listed above but they are not strictly speaking necessary. French doors have whats called an astragal (the strip on the passive door that overlaps the swing door) that unifies the two doors when closed. In addition the passive side will have a flush bolt top and bottom that drives a tab into the header and threshold.
You can arrange your closet door in this fashion but I've never seen it done that way for a residential closet, its just overkill. Most of them have twin ball catches on top (shown above) and nothing on the bottom, or a bi-fold style door aligner on the bottom that ensures the doors stay flush.
here's a link to a surprisingly complete and accurate retro fit of a door set like your describing.