doors
I have two doors at a 90-degree angle to one another and the doors hit each other when they are opened the full way.
I guess I could mount a door stop on one of the doors, but it seems kind of weird to put a doorstop on a door.
Another option might be to remount both the doors to either/both change the handedness or swing direction, but that would be a lot more work.
How can I keep these doors from hitting when opened?
This type of door is called a pocket door.
The first possibility is that the door has simply come off the track or the door has come unattached from the hanger that runs along the track. In that case you can figure out which side of the track it should be hung from and pick the door up and rehang it or attempt to fish the hanger out of the pocket and reattach it to the top of the door.
The more likely case, in my experience, is that the hardware is broken and will have to be replaced. Here is a good guide to replacing the hardware on a pocket door: http://www.ronhazelton.com/projects/how_to_repair_and_replace_a_pocket_door
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.
Best Answer
Another solution: Make one of them close to avoid clashing.
I reversed my kitchen door so it opens into the hallway instead of into the room. That means it can clash with the cloakroom door if that's left half open. I solved it by fitting a (very gentle, in fact modified with a weaker spring) automatic closer to the cloakroom door, so it doesn't stop half open.
Rehanging a door can be tricky but is perfectly doable with reasonable DIY skills. It's one of the simplest tasks for which you need a decent set of chisels (in fact it's the main reason I even have chisels). The job needs proper planning: when I did mine there was enough space on the new side of the frame to make it simple but had it been on a different (non-structural) wall it would have been much harder.