Use Retainers to store stuff on, you get two free and I think you can buy another couple (?), you can store a lot of gear on just two of them.
You can either just raise crafting and craft your own gear as needed, some you can store, others you can trade for company seals. Retainers are great for selling or raising them to a DPS class for hunting depending on gear you have. You can de - synthesize lower gear and raise level for later on as well. Or just spirit bound turn it into materia and get achievements.
I've always run in that problem myself and I do have purchased all the retainers you can get on the mog station, as well as using our old FC's company chest, so I just made couple alts and store it there. Granted if it's transferable otherwise, you will just have to throw stuff away that might be not as important (potions, crafting mats etc) I hope we will be getting a glamour book/log with 4.0 or at least an extreme increase of both armory and inventory and not just a couple of slots. Till then, you're out of luck :)