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?

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.

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.