Storing antimatter is an engineering nightmare. It's hard enough to store chemicals that react with lots of things, imagine storing something which reacts releasing insane energies with all matter. You have to confine it perfectly using external fields. In other words, not an easy task. (CERN has a trap that was demonstrated to have formed just a few antihydrogen atoms)
Not that we have any applications for it with the current production efficiency.
To my understanding, this isn't correct. Storing anti-matter isn't really that hard. The Tevatron would store anti-protons in the recycler for hours, and I'm sure if it was important we could engineer much more efficient mechanisms.
No, the hard part is creating the anti-matter in the first place and then cooling it down to a reasonable temperature at which it can be stored. All of our mechanisms for creating anti-matter generate anti-particles traveling at fantastic, highly relativistic speeds (and also consume much, much more energy to create then they contain in mass energy). Cooling them down without loosing them all to annihilation is very difficult. But once they're cool, it's not so bad.
Oh I'm sure you're right, didn't know that (my knowledge comes from years of exaggerated Scientific American articles :) ).
I suppose the cooling is electromagnetic, the whole process must be really difficult! Cool it while keeping those atoms from interacting from stray particles if the vacuum isn't perfect.
Perhaps more importantly, there's the messy consequence that those external fields only work on charged particles.
You could (in principle) keep positrons and antiprotons together in a very hot plasma under magnetic confinement, but every pair that binds together into a neutral hydrogen atom can immediately escape and annihilate - turning carefully-stored fuel into gamma rays...
This is a book from 1988 by a favorite author Robert L Forward (the worst author from which I regularly bought books in hardcover; I just read them for the physics).
Not that we have any applications for it with the current production efficiency.