I personally just leveled classes.

So instead of sitting with like 5 different sets of gear for differently leveled classes of the same type, I now just need 1 set per job.

Which leaves enough room for everything to fit into by Armoury. Though, weapons and rings become a bit of a squeeze due to me also carrying a couple of Relics and well rings are just awful (Fending, Aiming, Slaying, Casting, Healing, Gathering, Crafting... That's 14 rings right there. +2 for every other level you have sitting around...)

Otherwise I just get rid of most stuff.

With levels 50, 60 and 70 all giving a set of gear for (Most) classes, as well as the ease of purchasing a few sets between those levels from vendors, I don't really hoard any gear and when I do, it's just vendor bought stuff that I'm using to level up a particular role of class at the time.

For example, I recently finished leveling up all the Magical DPS classes, so I had level 50-68 gear stored away for SMN/RDM/BLM and focused on leveling them up and after finishing getting the last class up to 70 I could then get rid of the gear as I outleveled it.