To Increase Your Survivability in Combat
The only question to be found above was in your heading, so I'm going off of that: how to get the most AC and HP as a Bear Barbarian. But I'll also focus on making that AC and HP work for you: specifically in helping you take less damage in the first place, and give you some partial healing abilities.
The first advice I'd give for increasing your AC is to raise your Dexterity up to 20 ASAP. This will give you an AC of 10+5+5+2+2=24. (With your Unarmored Defense, and two shields). From there, you might want to see if you can grab some magic items like the Cloak of Protection, or Ring of Protection, both of which add +1 to your AC and saving throws. And naturally, a couple of Magic Shields couldn't hurt, as many of them raise your AC without requiring attunement (credit to daze413 for this point). But some of the best defense bonuses you can get come from multiclassing1.
Multiclassing:
Your Bear Barbarian's best trait to help your survivability is his resistance to all damage, when raging (other than psychic). But raging won't allow you to cast or concentrate on spells. So any further aids to your defense will be based on routes other than spellcasting.
For Being Hit Less/ Taking Less Damage from Hits:
Counterintuitively, you might want to take a several levels in Rogue or Monk. With your AC so high, you're unlikely to get hit by much other than spells. By taking up to level 7 in Monk or Rogue, you'll gain Evasion:
When you are subjected to an effect that allows you to make a Dexterity saving throw to take only half damage, you instead take no damage if you succeed on the saving throw, and only half damage if you fail. PHB p-96
Since Barbarians have advantage on Dexterity saving throws already (Danger Sense), this could add up to considerably reduced damage for you. Not to mention the the "half damage" guaranteed by this ability would stack with your "resistance" causing you to take a quarter damage overall, worst case scenario (if you are raging).
Rogues main combat advantage is their sneak attack, which you couldn't use unless your DM considers shields to be finesse weapons (unlikely). But you'd also get the 5th level Rogue ability of Uncanny Dodge, which would allow you to use your reaction to halve damage from one attack per round (with your Reaction) that manages to hit you (which would mean you'd take a quarter damage overall, since the Rogue's ability is not "resistance", and would stack with your current resistance to damage while Raging).
Monks, on the other hand, gain an ability to dodge as a Bonus Action at level 2, at the cost of a "Ki" point. If you took up to Monk level 7, you could do this seven times per short rest, which would be quite effective at keeping you from being hit. You'd also gain some control and damage ability from the Monk's "Stunning Strike", and healing ability from the Monk's "Wholeness of Body" (if you chose Way of the Open Hand as your Monastic Tradition).
Adding to your HP/ Being a Damage Sponge:
You've requested to "avoid magical classes". I don't know if you mean you want to avoid classes that require casting spells (which makes sense because of your Raging restrictions), or if you were opposed to classes that have spell slots overall. If it's not the latter, a good choice might be a Moon Druid.
Technically, transforming into an animal isn't a "spell", so you'd still be able to maintain a rage in a transformed form, and transform into (and out of) it as bonus actions while continuing to rage. This would give you all the HP of the beast you changed into, which would be effectively doubled since you will still take half damage while you Rage. Also, you'd still be able to use the Druid's spell slots to heal yourself while transformed.
Additionally, while you are transformed by Wild Shape, you can use a bonus action to expend one spell slot to regain 1d8 hit points per level of the spell slot expended. PHB p-66
Since this isn't a spell either, it's compatible with your Raging, allowing you to take half damage and still heal yourself.
The downside to this build is that the transformed shape is unlikely to be very high AC. Even using your "Unarmored Defense", most beasts still would have an AC of 15 or lower. Still, it's one of the highest survivability builds out there, and when the beast HP goes down you're back to your high AC self with full HP.
Personally, I'd recommend Barbarian 5, Monk 8 (ASIs in Dex), Moon Druid 2 for your level 15 character. It would provide a combination of incredible evasion (dodging most rounds) and a major HP buffer, both of which are replenish-able on a short rest. You'd only get 3 Rages per day, but that will allow you quite a bit of time where you are an unhittable, unstoppable steel-skinned damage sponge.
1 Small addendum to multiclassing: Barbarians gain the highest amount of hit points every level per class. So all of this multi-classing will cost you about 20 hit points to your regular HP, since Rogue, Monks, and Druids all earn 2 fewer hit points per level than Barbarians. But between the 74 hit points per short rest you're getting (from transforming into a Dire Wolf or Tiger twice, with 37 HP each), and the extremely increased damage reduction you'll get from Monk (near constant dodging and 0 damage from succeeding on an extremely common saving throw for which you have advantage), I firmly believe you're coming out ahead.
Best Answer
Too good when your offense doesn’t need hands
In theory, giving up a hand for more defense could be a fair trade—if you were losing offensive capability in the exchange. But as Dan B’s answer details, that just isn’t the case for far too many classes, particularly one your character would be likely to gravitate towards naturally, the cleric.
But maybe we could avoid that
Maybe we can avoid the problems by giving up on the AC bonus of the second shield—but still having it be a meaningful part of the character.
Shield as a weapon
The rules do not allow cover bashing things with a “real” shield. There is precedent, however—in the Monster Manual—for shields as weapons. The lizardfolk has an attack called “spiked shield” that deals 1d6+Str damage. At the same time, the lizardfolk’s 15 AC is described as being due to “natural armor, shield.” The lizardfolk shaman on the next page has “AC 13 (natural armor),” and it does not have a shield in its statblock. So the implication here is that the spiked shield provides the usual +2 AC bonus that a regular shield does, as well as being available for an attack.
You can’t do that as a player character—there is no spiked shield in the rules, using a shield to do damage forces you to treat it as an improvised weapon (which makes it difficult to get proficiency and limits its available damage potential), and so on. However, what you might do—with DM permission—is treat the spiked shield as a weapon only. After all, the lizardfolk’s ability to use the spiked shield at the same time as their regular weaponry is due to their multiattack ability—and you don’t have that.
So you could ask to have a “shield” that is designed to be used as a weapon, with the stats of, say, a mace or morningstar, and no bonus to AC. Or maybe it could do 1d6 piercing or 1d4 bludgeoning, i.e. a die size smaller than those, but be resistant to disarming. Then you just use it as a regular weapon, but it pays the appropriate homage to your god.
A homebrew feat or fighting style might be appropriate here
If the cost of using two shields is just “gives up a free hand,” then it’s not balanced because you can do too much without needing hands. But if the cost is more than that—say, you need to get a fighting style from fighter or paladin, or you need to spend a feat—it might be more reasonable.
It’s too far out of scope here to try homebrewing something, but the Defense Fighting Style or the Dual Wielder feat might be well worth considering as a starting point. Consider that someone wearing armor, with a shield and a “fake weapon–shield,” could take the Defense Fighting Style for a +2 bonus to AC from the shield and a +1 bonus to AC from the fighting style—you could easily just flavor that as your “fake weapon—shield” being half as good as a “real” shield for defensive purposes. If you had a fighting style that required two shields—a stricter requirement than Defensive’s simple “wear armor” requirement—you might be able to justify a better benefit than that. A feat might be able to go even farther.
You can ask your DM for help working out the details and coming up with something acceptable to you both—and you can ask about those details, once you’ve decided on them, here, and we can judge whether or not you’ve hit the right sweet spot.