Let's do a racial breakdown, just because. I mean, why not? It's a nice Friday, and I'm finally off my dayjob.
Obviously there's always the risk that I've messed up in the table, but here goes.
https://pasteboard.co/o23tHfthDBxX.pngSo let's consider things. Since we're talking Barbarian class, all the racial weapon and armour proficiencies are pretty much irrelevant. Keeping that in mind...
Dragonborn are well over 6 foot tall and easily weigh 250 lbs, so of course they're not really any better at the whole barbarian thing than your average garden gnome. They have 1 resistance and a so-so once per rest breath weapon, presumably. Not horrible but they just don't have a lot that synergizes with the Barbarian class.
Shield dwarfs have 40' darkvision, resistance vs poison, and advantage vs poison saves. Gold dwarfs have that and +1 health per level. Both have a fairly significant penalty to movement speed. Not my first or second choice, but if you want to be a dwarf barb then do pick the goldies, because shieldies are objectively inferior, full stop.
Elves are interesting. Wood elf barbarians get 35' movement speed, 40' darkvision, resistance vs charm, immunity vs sleep, and proficiency to perception and stealth checks. High elves trade movement speed for a wizard cantrip, though none of them really help barbarians that much. Drow get better darkvision but have to choose Dancing Lights, a concentration cantrip that illuminates dark areas, which is actually kind of bad.
Gith don't really have anything going for them as a martial class except for their psionic spellcasting. Cast jump before raging and you've got all the mobility you could ever want. But that's the only bonus they get. No resistances, no advantages versus anything, no extra damage. But that mobility they get is immense.
Gnomes are sort of interesting. They get inherent proficiency in Int, Cha, Wis saves, which complement Barbarian Str, Con save proficiency nicely. They get Darkvision too, which isn't bad. Deep gnomes are masters of stealth. Forest gnomes get a single cast of speak with animals. And rock gnomes get history proficiency x2. Downside, all gnomes have 25' movement speed. Not the best, but a greataxe swung by a gnome leads to exactly the same headache as one swung by a dragonborn or a gith.
Half-Elves are just flat out inferior Elves, so in that sense it makes sense to skip them. Worth nothing that Half-woodies are still significantly better than non-woodies for barbarity because of their movement speed.
Halflings are an amusing choice for a Barbarian, but they're not that well suited. Rerolling d20 rolls of 1 is nice but it's probably better for skill monkeys. Lightfoots get stealth proficiency, Stouthearts get advantage versus frightened effect. Neither get darkvision and both have 25' movement speed, which kind of sucks.
Half-orcs have darkvision, standard movement speed, and two unique abilities, both of which work well with a martial tank class. Relentless Endurance lets them stay at 1 health if they're reduced to 0 without getting killed outright. Savage Attacks lets you deal your weapon die as additional damage when you crit. Pretty good.
Humans get +20 bls of carry weight and regular movement speed. Garbage. Skip.
Tieflings get regular movement speed, darkvision, and resistance to fire. And they get some spellcasting that doesn't synergize at all with Barbarian. Not horrible, not exactly great either.
So, to summarize, my tentative ranking would be as follows.
S-tier: Half-orc, Wood Elf
A-tier: Woodie Half-elf, Gith
B-tier: Tieflings, Gold dwarf
C-tier: Dragonborn, deep gnomes, Drow elf, High elf,
D-tier: Shield dwarf, forest gnome, rock gnome, Drow half-elf, high half-elf
F-tier: Human, Halflings
Note that all reasonable disclaimers apply. The game is not released yet, this is not playtested, and speculative guesswork is ultimately just that.