For example, the Path of the Zealot Barbarian subclass at 14th level gets the Rage Beyond Death feature (XGtE, p. 11), which prevents you from dying while raging:
Beginning at 14th level, the divine power that fuels your rage allows you to shrug off fatal blows.
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.
What other mechanics available to players are there that either postpone or cancel death?
Considering your example, I'll interpret "counteract death" as "postpone dying" without the use of healing mechanics that the game offers.
Ways to counteract death by yourself
Picking half-orc as race, from the Player's Handbook (p. 40-41), grants:
When picking barbarian from the Player's Handbook (p. 46-50), you'll get:
Subclass: Way of the Long Death (Monk)
Monks that follow the Way of the Long Death, from the Sword Coast Adventurer's Guide (p. 130-131), gain:
Subclass: Samurai (Fighter)
As pointed out by Blckknght in a comment, the Samurai fighter subclass, from Xanathar's Guide to Everything (p. 31), grants the player:
During that extra turn, you could possibly take a potion (as an action) and/or use Second Wind to carry on fighting (or retreating).
Subclass: Oath of the Ancients (Paladin)
As mentioned by Stackstuck, the Oath of the Ancients paladin subclass, from the Player's Handbook (p. 87), gives a character:
Subclass: Shadow Magic (Sorcerer)
As noted by Davo, the Shadow Magic sorcerer subclass, from Xanathar's Guide to Everything (p. 50-51), gives you:
Spell: Death Ward (Cleric, Paladin)
As suggested by Miniman, the 4th-level abjuration spell death ward from the Player's Handbook (p. 230), grants a similar ability:
If you target yourself with this spell, you don't actually require assistance of another character.
Ways to counteract death (saving throws), with assistance
I'm not certain if the following options fit what you're looking for. But as suggested by commenters, I'm including these because they also "postpone death without the use of healing", by stabilizing the target:
Spell: Spare the Dying (Cleric, Warlock)
As mentioned by SirTechSpec, the cantrip spare the dying from the Player's Handbook, also allows for a sort of similar way to counter death (without healing):
If spare the dying counts, the Medicine skill from the Player's Handbook, will also suffice in countering death without the use of healing:
Depending on the DM, you might only attempt a Medicine check if you're proficient with it. But as far as I know, by RAW, any player character can attempt to stabilize another character.
Adventuring Gear: Healer's Kit
As L0neGamer points out, a Healer's Kit can have a similar mechanic to a Medicine check.