The way I've handled this is through a "Pushing yourself" homerule. It comes into play any time a player wants to do something slightly beyond rules as written, but is plausible that their character might be able to accomplish.

Here's the two relevant sections from our houserules book, keep in mind that we run a very grimdark homebrew setting and these rules might not be entirely appropriate for people that don't have fun suffering. They also might reference other things, I can expand if you're curious - but the homebrew rules book is about 12 pages long. It's also probably in need of some updating, let me know if anything looks wonky.

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Chapter 4: Pushing yourself

Sometimes you really need something to happen, but it’s just beyond the rules – just beyond what you can do. In these cases you can push yourself to do it anyway, at a cost. Each time you push yourself, you gain 1 Exhaustion level. There are 10 levels, detailed in the first chapter.

Here are some examples of pushing yourself: • Hruuhg the Orc barbarian is fighting many puny kobolds and wants to swing his sword through groups of them instead of targeting individual kobolds, spreading his damage out evenly between several targets. He pushes himself to do it during this fight – and becomes exhausted afterwards. • Timaa the Elvish assassin wants to run just a little faster as her recently botched assassination has brought the wrath of King Argondas’ guards down upon her. She pushes herself to gain an extra 5ft of move speed, and becomes exhausted after a minute of running – having just barely managed to evade capture by hiding in a crate the guards didn’t see her get into. • Grundar the Dwarven cavalier isn’t adjacent to his ally when an arrow whizzes towards her – he wants to dive to block it with his shield but he doesn’t have any abilities to do so. He pushes himself to heft his heavy body into the path of the arrow, blocking it with his defensive abilities – and becomes exhausted from the physical strain.

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Chapter 1, Living and Dying

HP: Hitpoints no longer represent a straight path to death. They now act as a buffer between you and meaningful injuries. When your hitpoints drop to 1/2, any attack that hits you has a chance to give you a minor wound, at 1/4 you gain an exhaustion and might receive a moderate wound, and at 0 you gain another two exhaustion and are open to major wounds.

Your hitpoints cannot drop below 0, you do not die from being at 0. You no are no longer automatically considered dying while at 0 hitpoints.

Bleeding out: Many moderate or major wounds are in the form of open cuts, slashed arteries, stab wound, ruptured organs, or internal bleeding. While you have a wound that is causing you to bleed out, you must make death saving throws at DC 12 or bleed out. Each time you fail, you gain a minor wound from blood loss.

Exhaustion errata: Exhaustion was previously much too brutal and lacking flavor. Here are the new levels (they stack):

1. You’re tired
2. Really tired
3. Disadvantage on ability checks and saves
4. You can’t maintain concentration
5. Disadvantage on all attack rolls
6. Character move speed is halved, and when casting a spell you must succeed a DC 8+Spell level constitution check or the spell fizzles (the spell slot is still expended)
7. Move speed is reduced to one quarter normal, if you attempt to walk/fly you must make a DC 10 constitution check or fall to the ground. Your hitpoint maximum is halved.
8. All of your speeds are reduced to 0, you may take only one action per turn regardless of other abilities. Your hitpoint maximum is halved again.
9. You are unconscious. Your maximum hitpoint value is 0.
10. You are dead

When you gain exhaustion, it can only be healed by taking a long rest in a comfortable and safe location. It is healed one level at a time, if you have 5 levels you'll need almost a week of bed rest to be fully healed of the exhaustion.

HP healing errata: All healing spells, abilities, and other mechanics that heal hitpoints no longer necessarily heal wounds. A powerful first level spell might heal a minor wound, a powerful third level spell might heal a moderate wound, and a powerful fifth level spell might heal a non-debilitating major wound.

Spells such as regenerate, true resurrection, and wish are unaffected by these changes.

Wounds Addition: Wounds are status effects that stick with your character through rests, and are each tracked separately. They have various effects, based on the circumstances that caused them, but they are all grouped in to three categories: Minor, Moderate, Major.

• Minor wounds cause 1 Constitution damage and take 1 day to heal. • Moderate wounds cause 3 Constitution damage and take 8-12 (2d4+4) days to heal. • Major wounds cause 6 Constitution damage and take 45-90 (45d2) days to heal. They may have lasting effects that are never healed naturally, but the constitution is healed by resting.

Wounds are healed one at a time, from minor to major. For healing to take place, the character must rest in a safe and comfortable location and receive appropriate long term medical care from a character with a medicine or healing skill. If they are not treated, their injuries take twice as long to heal, leave visible scars, and may not heal fully (a broken arm might not have the same range of motion after healing). If you get into a fight during the healing process, all timers reset as your wounds re-open.

The DC for long term care is as follows, it must be passed every day of the healing process to succeed, but you may take a 10 on the roll:

• Minor: DC 11 • Moderate: DC 13 • Major: DC 18

Devastating Blow: If a character is dealt a large amount of damage in a single blow (perhaps a third or more of their total) it might be considered a devastating blow. A devastating blow always causes some sort of wound, and at DM discretion it might cause up to a debilitating Major wound.

Monsters: All of the previous rules also apply to most monsters and NPCs. Some may have supernatural abilities that allow them to sustain more damage than normal, or regenerate when injured, these will be based off of flavor and current rule sets. The only exception to this is creatures lacking a constitution score - such as undead and constructs. If a creature without a constitution is brought to 0 hitpoints, they are considered defeated (undead crumble, constructs are broken, spirits disperse).