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
Race: half-orc
Picking half-orc as race, from the Player's Handbook (p. 40-41), grants:
Relentless Endurance.
When you are reduced to 0 hit points but not killed outright, you can drop to 1 hit point instead. You can’t use this feature again until you finish a long rest.
Class: Barbarian
When picking barbarian from the Player's Handbook (p. 46-50), you'll get:
Relentless Rage. Starting at 11th level, your rage can keep you fighting despite grievous wounds. If you drop to 0 hit points while you’re raging and don’t die outright, you can make a DC 10 Constitution saving throw. If you succeed, you drop to 1 hit point instead.
Each time you use this feature after the first, the DC increases by 5. When you finish a short or long rest, the DC resets to 10.
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:
Mastery of Death.
Beginning at 11th level, you use your familiarity with death to escape its grasp. When you are reduced to 0 hit points, you can expend 1 ki point (no action required) to have 1 hit point instead.
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:
Strength Before Death. Starting at 18th level, your fighting spirit can delay the grasp of death. If you take damage that reduces you to 0 hit points, you can use your reaction to delay falling unconscious, and you can immediately take an extra turn. While you have 0 hit points during that extra turn, taking damage causes death saving throw failures as normal, and three death saving throw failures can still kill you. When the extra turn ends, you fall unconscious if you still have 0 hit points.
Once you use this feature, you can’t use it again until you finish a long rest.
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:
Undying Sentinel. Starting at 15th level, when you are reduced to 0 hit points and are not killed outright, you can choose to drop to 1 hit point instead. Once you use this ability, you can't use it again until you finish a long rest.
Additionally, you suffer none of the drawbacks of old age, and you can't be aged magically.
Subclass: Shadow Magic (Sorcerer)
As noted by Davo, the Shadow Magic sorcerer subclass, from Xanathar's Guide to Everything (p. 50-51), gives you:
Strength of the Grave. Starting at 1st level, your existence in a twilight state between life and death makes you difficult to defeat. When damage reduces you to 0 hit points, you can make a Charisma saving throw (DC 5 + the damage taken). On a success, you instead drop to 1 hit point. You can't use this feature if you are reduced to 0 hit points by radiant damage or by a critical hit.
After the saving throw succeeds, you can't use this feature again until you finish a long rest.
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:
You touch a creature and grant it a measure of protection from death.
The first time the target would drop to 0 hit points as a result of taking damage, the target instead drops to 1 hit point, and the spell ends.
If the spell is still in effect when the target is subjected to an effect that would kill it instantaneously without dealing damage, that effect is instead negated against the target, and the spell ends.
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:
The best way to save a creature with 0 hit points is to heal it. If healing is unavailable, the creature can at least be stabilized so that it isn't killed by a failed death saving throw.
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):
You touch a living creature that has 0 hit points. The creature becomes stable. This spell has no effect on undead or constructs.
Skill: Medicine
If spare the dying counts, the Medicine skill from the Player's Handbook, will also suffice in countering death without the use of healing:
A Wisdom (Medicine) check lets you try to stabilize a dying companion or diagnose an illness.
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.
This kit is a leather pouch containing bandages, salves, and splints. The kit has ten uses. As an action, you can expend one use of the kit to stabilize a creature that has 0 hit points, without needing to make a Wisdom (Medicine) check.
