Elf: Base increase to Dexterity is of moderate use.
Drow: Sunlight vulnerability is a huge drawback.
EladrinMToF: Fey step is very good, but otherwise this race isn’t great for Barbarians.
High Elf: No useful abilities for Barbarian
Sea ElfMToF: The Constitution increase is the only useful trait.
Shadar-KaiMToF: Good for Dex-based Barbarians, and the teleport ability is great.
Wood Elf: Wisdom and a bit of extra movement.
FirbolgVGtM: Contrary to Barbarian philosophy, the Firbolg is tricky and sneaky.
GenasiEEPC: Bonus to Constitution, plus some nice perks in each subrace.
Air: A Dexterity increase and not needing to breathe are nice perks but not incredibly powerful.
Earth: Strength increase and being able to ignore difficult terrain makes you dangerous.
Fire: Intelligence doesn’t help you at all.
Water: Wisdom increase, and the ability to breathe underwater
GithMToF: Githyanki are very strong for Barbarian, but Githzerai are the worst.
Githyanki: Strength increase, proficiencies, and some spell-like abilities is a great combo package.
Githzerai: This race has nothing helpful for Barbarians.
Gnome: No Gnome is a good Barbarian
Deep (Svirfneblin)EEPC/SCAG: Too sneaky to get the point across.
Forest: Nothing useful here.
Rock: The Constitution bonus is helpful, but the other abilities are off-theme.
GoblinVGtM: The mobility options are compelling, but otherwise not very good.
GoliathVGtM/EEPC: This race was meant to be a Barbarian. Strength, Constitution, and the ability to negate some damage make this a particularly potent candidate for Barbarian.
Half-Elf: Half Elves make a half-decent candidate for any race because of their versatility.
Standard: Put your ability score increases into Strength and Constitution and pick up two skills on your list.
AquaticSACG: Only good for an aquatic campaign.
DrowSCAG: The spells are helpful, but you’re going to be focusing on dealing damage rather than messing around with magic. Faerie Fire gives you a way to deal with invisible foes, which is one of the ways you can be vexed in combat.
Moon/SunSCAG: Take Green Flame Blade or Booming Blade as your cantrip. Otherwise, not very useful.
WoodSCAG: The extra movement is nice, but otherwise useless.
Half-Orc: Another race that was born to be a Barbarian. Strength and Constitution increases, Darkvision, and Intimidate are all the makings of a good Barbarian. Savage Attacks combined with Brutal Critical ensure a wicked amount of damage on a critical hit. Likewise, Relentless Rage and Ruthless Endurance give you tons of chances to stay alive in a fight to the death.
Halfling: Small size is an issue.
Lightfoot: Nothing good for Barbarian.
GhostwiseSCAG: Nothing good for Barbarian.
Stout: The Constitution increase makes this the only Halfling subrace worth considering.
HobgoblinVGtM: Constitution boost and Save Face are good