At 20th level, the Path of the Zealot barbarian (Xanathar's Guide to Everything, p. 11) can rage indefinitely by virtue of starting a new rage just prior to their initial rage ending. This is confirmed by this question: Can I start a new rage before the previous one ends?
At 14th level, they also picked up the Rage Beyond Death feature, which lets them stay active at 0 hit points and regardless of how many death saves they've failed:
While you’re raging, having 0 hit points doesn’t knock you unconscious. You still must make death saving throws, and you suffer the normal effects of taking damage while at 0 hit points. However, if you would die due to failing death saving throws, you don’t die until your rage ends, and you die then only if you still have 0 hit points.
At 15th level, barbarians gain the Persistent Rage feature, meaning their rage ends only if they fall unconscious or choose to end it.
Given that the Barbarian can constantly remain in a state of rage, how can I ever reliably kill them?
"Reliably" meaning they won't get a save to negate the effect, since they would be able to constantly cycle their Fanatical Focus feature to ensure rerolls on saves.