Well i think it will take a very inexperienced player to not to start with a Dwarf that can have 18 str and 18 con.
Thats the first thing . Well i would still go for eathel's hair and pump up my con take human or dwarf. And take dual wield or polarm master. But polarm isn't implemented in the game yet.
Second the key to win with a barbarian is to exhaust the paladin to out heal the damage with the Bear Barbarian.
But lets take that bad build frenzy berserker.
Currently from patch 8 prone is bugged. But if it would work that berserker can throw on the paladin a weapon and the paladin whould knoked down to prone and prone gives advantage w/o reckless attack. At lvl 5 barbarian's gain extra movement. Vs the half moving paladin .
Yes the paladin can throw a weapon to but there is a huge difference between 1d6+4+2x3 vs 1d3+2x2.
I want to point out something else the dps and survival race isnt favouring both paladin and barbarian.
The winner is Larian's Beast Master Ranger. With 5 attacks unlimited animal companion / frenzy raging boar with dex save prone . He have always advantage and hunters mark is unlimited. And can also dump dexterity to use heavy armor.
Currently nothing can beat the beast master with great weapon master + hunters mark.