I have played a barbarian tank in a previous campaign and I am currently playing a oath of vengeance paladin in the current campaign.
First about barbarians. They can be hands down the best traditional tank out there. If you go path of the totem warrior and take all of the bear powers you will effectively double your hp pool while also inflicting disadvantage on any enemy who attacks anyone other than you in melee.
That gives creatures one reason to attack you. Combine that with sentinel and now you are very sticky and can stop enemies in their tracks with an OA, or attack them if they try and attack your buddy with disadvantage. Also you can just constantly give yourself advantage every round, while giving everyone else advantage who wants to attack you. As a tank this is essentially a taunt and given your damage resistance, not really a problem.
That is four powerful reasons for anything in melee range to attack you over your friends and you haven't spent an action yet.
Another great feat for barbarian's is shield master. Now when the big bad dragon blasts you with it's breath attack, you already had advantage on the dex save, you can add your shield modifier to that save, and if you pass you take 0 damage or if you fail you take 1/4 damage.
Having high strength is sort of the barbarian's thing, picking athletics as one of your proficiencies is highly advisable. Getting advantage on all strength checks while raging is really great. Mix it all together and you are a grappling monster. The only thing you can do to make it even better is to take a single level of rogue and gain expertise in athletics. Grappling and shield master work very well together. Grapple, and then shove prone as a bonus action.
So after all of that you have one of the most durable fighters in the game who can lock down two to three creatures each round. You wont be doing much for damage, but that is not your job. You are a walking choke point for the rest of the party to focus their attacks on. (I wish I had found that shield that cursed you to always become the target of ranged attacks, would have been beautiful.)
Secondly the paladin build I am doing right now is Oath of vengeance with polearm master and sentinel. This is a very different tank. You can still stop creatures in their tracks, sometimes where they are incapable of attacking back. You can buff the party, heal, whip out some clutch damage when you smite on top of a critical, and give people a reason to stay near you with your protective aura, which works great for the reach enabled sentinel powered OA's.
The other great thing about vengeance paladins is their mobility and battlefield control. Abjure enemy takes a melee monster out of the fight. Compelled duel can lead a monster straight into your sentinel OA. Relentless avenger when coupled with all of the OA's you will be swinging lets you jump around the field like a monk or rogue. Haste is just pure win on top of everything else and works great with dumping more smite damage in one crazy round.
To be honest though... the best tank in the game is a wizard, or warlock, or sorcerer or bard... or anyone that picks a bunch of crowd control spells. If you focus heavily on reducing the onslaught of enemies down to a trickle... it doesn't matter what the odds are, you are going to wipe the table clean.
Wildshape druids are an honorable mention as they are tougher than nails with all of that free hp. But I find they suffer in the stickyness department. Even with sentinel, their +to hit in their wildshape tends to lag behind their non shape shifted counterparts. If they summon a bunch of creatures and then shape shift AND take some feats like resilient constitution and warcaster to maintain concentration, then they can basically create a wall of fur between the enemy and the party. A pack of wolves is amazing at knocking people on their butts and a pack of pixies can put the world to sleep.