How do I store pens so they won't leak on one another?

This strikes me as solving the wrong problem- you should get some pens that don’t leak. I’m a fan of the Uni-ball Jetstream, but there are lots of good options. 

If you’re talking about storing fountain pens, there’s an instructive video by Goulet Pens. For travel, try storing them nib up. For fancy storage, there’s a host of options: Dudek, Aston, Visconti, etc.