Not only that, but even games without a sub model (like GW2) provide the opportunity to purchase extra bag and bank room every time they release new content.
So, even if you don't sub to ESO+, you should be able to buy it separately here, too.
Here's what I do:
Saving: Storage in MMOs isn't so you can save everything you come across like single player games. If you really want to save something "has potential" or "might be good one day" then you should only save the jewelry (if gold, maybe purple) and weapons. Any body pieces are easy to re-farm. Never save craftable stuff after it's been a few patches (maybe exception for jewelry), if it's good years down the road, re-craft it IMO. Never save zone drops you are not going to use for something immediately, those items are almost always on guild traders regularly, especially if they are from a good set (exception for picking a weapon of perfect trait I could see). For boss heads, I'll usually save one of each size type and if I want to cut that down I'll cut heads over shoulders since I can refarm heads easily and should cost currency to re-aquire.
Anything not gear sets like recipes and such either get used or sold immediately. If you have a lot of stored up items to move, then start moving them. Price the stuff to go (you don't have to lose your shirt on them, just price them at a going rate and they will go, especially if you use TTC and actually make sure your items get posted to the site). If you have 300 recipes and motifs to sell, you may only get 30 slots, but it would only take a week to move probably 100+ of that if you prices it competitively.
For storage, I keep 2 large chests of undaunted pieces, 1 large chest of extra tanking gear that isn't worth keeping in bank, 1 large chest of weapons, 2 small chests of jewelry, 1 small chest of extra gear and 1 small chest of extra pvp gear. All of those are about 80% full. That's where I tend to keep the "just in case" it's useful again or finally stuff and why it's mostly jewelry and weapons.
Organization: Housing can eat up a lot of space, but I have a few homes that I don't use for decorating and use for storage. I separate these homes into one for lighting, one for basic furniture and another for specialty furnishings. You can also unload furnishings that are not bound by selling them in guild traders.
Also on organizing gear- I keep the most common sets on each of my characters. For example, I have a tank of each class, and each tank is carrying the typical Ebon/Yoln/Alkosh + monster sets gear and a gear set for questing. All other sets that have niche uses for tanking like say Eternal Yokeda or Defiled Dragon, etc. I keep one set in the bank that any tank could pull out to use for a particular need, then put it back when done. And of those sets I typically keep only the jewelry/weapons (because I'd drop alkosh for them). This is the same for DPS, though I don't have nearly as many 'just in case' type set ups for DPS, usually what's good is good. And I keep a few general use things in the bank like extra soul gems and repair kits, the different food and pots for daily writs, some of the rewards we get like crown food.
As a player that does pve and pvp, housing, raiding, questing etc. I will say it is possible to keep a good inventory, but you have to stay on top of it or it becomes a job trying to get it fixed. For me things stay at about -100 items on each character that way they have 100+ slots for whatever they pick up, bank is normally 100+ short, trying to keep it under 360 for throwing something in the bank for crafter to deal with or move between characters, and chests kept at about 80%. This gives me some buffer for being lazy for a time or farming or whatever might cause an inventory bubble. If you have any extra character you want to use as mules, you'll be in an even better spot than I am.