To increase your ability to survive in combat
The only question that was found above was in its header, so I get out of that: how to get the most AC and HP as Bear Barbarian. But I'll also focus on making that AC and HP work for you – specifically to help you take less damage in the first place, and give you some partial healing skills.
The first tip I would give to increase your CA is raise your skill up to 20 as soon as possible. This will give you an AC of 10 + 5 + 5 + 2 + 2 = 24. (With your Defense without armor and two shields). From there, you may want to see if you can grab some magical items like the Protection layeror Protection ring, which add +1 to your AC and saving throws. And naturally, a pair of Magic Shields cannot deal damage, as many of them increase their AC without requiring tuning (credit to daze413 for this point). But some of the best defense bonuses you can get come from multiclassesone.
The best trait of your barbarian bear to help him survive is his resistance to all damage, when he is furious (other than psychic). But fury will not allow you to cast or focus on spells. Therefore, any additional help for your defense will be based on routes other than casting spells.
For being hit less / taking less damage from hits:
Counterintuitively, you may want to take multiple levels in Rogue or Monk. With your AC so high, you're unlikely to get hit much harder than spells. By taking up Level 7 in Monk or Rogue, you will win Evasion:
When you're subject to an effect that allows you to make a Dexterity saving throw to take only half the damage, instead you take no damage if you succeed on the saving throw, and only half damage if you miss. PHB p-96
Since the barbarians already have the advantage on saving throws of Dexterity (Sense of Danger), this could add considerably less damage to you. Not to mention, the "half damage" guaranteed by this ability would build up with your "stamina", causing you to take a quarter damage overall, in the worst case (if you're furious).
The rogues' main combat advantage is their sneak attack, which you cannot use unless your DM considers shields to be fine weapons (unlikely). But you would also get Uncanny Dodge's fifth level Rogue ability, which would allow you to use your reaction to halve the damage of one attack per round (with your Reaction) that manages to hit you (which would mean you would take a quarter damage in general, since the Rogue's ability is not "resistance", and would accumulate with your current resistance to damage while Rage).
Monks, on the other hand, gain the ability to dodge as a Bonus Action at level 2, at the cost of a "Ki" point. If you reached Monk level 7, you could do it seven times for a short break, which would be quite effective in preventing you from getting hit. You would also gain some control and damage ability with the Monk's "Stunning Strike", and the Monk's "Totality of Body" healing ability (if you choose the Open Hand Path as your Monastic Tradition).
Adding to your HP / Being a Damage Sponge:
You have requested "avoid magic classes". I don't know if it means that you want to avoid classes that require spellcasting (which makes sense due to your Raging restrictions), or if you opposed classes that have spell slots in general. If not the latter, a good option could be Moon druid.
Technically, transforming into an animal is not a "spell," so you can still hold an anger in a transformed form and transform into (and out of) as bonus actions while continuing the anger. This would give you all the HP of the beast you changed into, which would effectively double as you will still take half damage while raging. Additionally, you can still use the druid's spell slots to heal yourself while transformed.
Also, while Wild Shape transforms you, you can use a bonus action to spend a spell slot to recover 1d8 hit points per level of the spell slot spent. PHB p-66
Nor is it a spell, it is compatible with your Raging, allowing you to take half the damage and heal yourself.
The disadvantage of this construction is that the transformed form is unlikely to be very high AC. Even using your "Defense Without Armor", most beasts would still have a CA of 15 or less. Still, it's one of the tallest survival constructions out there, and when the HP beast falls, you return to your high AC with full HP.
I personally recommend Barbarian 5, Monk 8 (ASIs on Dex), Moon Druid 2 for your level 15 character. It would provide a combination of incredible evasion (dodging most rounds) and a great HP damper, which can be replenished in a short break. You would only get 3 rage per day, but that will give you plenty of time where you are an unstoppable and unstoppable steel skin damage sponge.
one Small multi-class annex: Barbarians get the most hit points in each level per class. Therefore, all of these multiple classes will cost approximately 20 hit points to their normal HP, as Rogue, Monks, and Druids gain 2 less hit points per level than the Barbarians. But between the 74 short rest hit points you get (transforming into a Dire Wolf or Dire Tiger twice, at 37 HP each), and the extremely increased damage reduction you'll get from Monk (dodge almost constantly and 0 damage from succeeding on an extremely common save shot for which you have an advantage), I firmly believe that you will get ahead.