So, a lot of this is probably going to be kind of "well no kidding" for people who played this game since ARR, but for those who are new and leveling multiple jobs and/or maintaining a glamour backlog...
1) Old dungeon sets from past expansions can be safely turned in for Grand Company Seals or sold off. It's so easy to get those armor pieces by just running the job you want them for that there is no point in stuffing them in your inventory.
2) Old armor sets from 2 expansions or more ago can be safely turned in for Grand Company Seals IF they are from 50/60/70 dungeons or old savage raid tiers. They are pretty easy to get back, though the one potential exception is the Leg armor since getting a group together for those parts can be a pain.
3) Poetics gear can be turned in safely for Grand Company seals or sold to NPCs. This can be argued a bit since some early sets get outdated faster than one could reasonably purchase them back, but for the most part they are not worth holding onto if not currently in use.
4) Crafting and Gathering materials from older expansions, unless they are from timed nodes, are not worth holding onto and should be sold or tossed out. You'll end up burning multiple retainers trying to keep a convenient backlog of items on hand and it just isn't worth it.
Now for the painful stuff...
5) If the item is from the cash shop, you should store it.
6) Veteran Rewards and original AF gear can be stored in the armoire. Seasonal items may or may NOT be stored in the armoire. So for the most part, store what you can in the armoire and then either stow or toss the remainder.
7) Housing items can be stored in your housing storage, but keep in mind that you'll want to avoid hording common items and sell them off. Seasonal items from events generally have to be stored permanently or repurchased from the cash shop.
8) Fantasia's, level boosts, and other account items must be stored, so plan on it.
9) Generally, crafting and gathering gear used for leveling is highly valued in the 60-89 bracket, so if you are planning on leveling multiple crafts, you'll have to hang onto that stuff. HQ crafting gear is one of the big sources of income ironically for other people participating in the gathering and crafting arena.
10) Consumable items unique to event areas like Eureka are generally more powerful than the common alchemical items, so they usually need to be stored as they are typically rewards for doing fates or other activities. However, you should absolutely clean out old potions and ethers as only the highest grade matters, and both potions and ethers are useless at max level.
11) If an item comes from Eureka or one of the major grind areas, you'll probably need to store it due to the time needed to acquire the item. This includes relic weapons, rewards from completing the deepest depths of deep dungeons, Eureka and Bazja, and Ultimates.