Also, are you supposed to hold your elbows in that position for long periods of time? I feel like there should be an extension to rest the forearms or wrists.
Resting the wrist or elbow on some sort of extension is what causes the strain. The trick is building the strength to hold up your hands and arms (maybe take up piano ;) ) without becoming tense in your shoulders and back. All of this follows from having a good posture.
If you're standing with proper posture, your upper arms are nicely aligned with the body, and your muscles should easily support the minimal weight of your forearms.
Edit: incidentally, as I'm experimenting with a standing desk, I have a stool which is about a foot too low to stand on. I do, however, find myself using it to accomplish all sorts of odd leg positions and stretches.
Keep in mind that you aren't typing or mousing all the time. In a standing position, I find that I'm much more likely to drop my hands by my sides, or cross my arms when thinking, than I am when sitting at a desk.