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.
![domed stop](https://i.stack.imgur.com/900Su.jpg)
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).
![mag latch](https://i.stack.imgur.com/bHzbP.jpg)
As an alternative, you could use a cabinet latch at the top.
![cabinet latch](https://i.stack.imgur.com/g99Dc.jpg)
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.
![button latch](https://i.stack.imgur.com/Iz5Wa.jpg)
Handles
In each of these cases, you would use dummy handles or pulls to open the doors.
You don't. Either stop the baseboard at the corner and live with the gap, or frame the opening with baseboard or molding and overhang the edge.
Notice how in this picture you can't see the metal track at the top, and you also can't see the gap between the outer left/right edges of the doors and the wall. That's because the molding overhangs the wall by 1/4 to 1/2 inch.
![enter image description here](https://i.stack.imgur.com/G6ENe.jpg)
Best Answer
Sliding doors may have a track at the bottom if they are doors riding in bottom side wheels. In this case definitely install the track first and fit up the doors. Carpet tack strip will be installed on either side of the track. Doors can be removed during carpet installation.
If the sliders are ones with a top hanger track that the doors hang from on rollers then there will be a center guide bracket that keeps the bottom of the doors aligned. With these you generally want to install carpet first as it will be very difficult to access inside the closet to fit the carpet through the opening with two hanging doors in the way. It will also be possible to better gauge the trim length for the doors once the carpet is in place.
There will be a decision point regarding the sliding door center guide on the floor. The best way to deal with that is to install a small square of plywood right where the guide will be located. This should be about the thickness of the padding. The padding would be cut around this block and carpet laid over the top. The guide will be installed with its screws going through the carpet and into the block below. The purpose of the block is so the guide does not compress the carpet all the way to the floor.