I had a lot of odd jobs to do around the house today, so I wasn't able to devote much time to the storage cabinet, but I did manage to fashion a couple more doors and affix the requisite hardware to them and get them mounted to the frame.
The fit was a little tight, but I'll be fixing that later this week. Everything works fine, though.
And here's a close-up of the door catches that I've installed. They're cheap. They work. That's all I need.