I wanted to check my books before replying to this. Page 293 of the 4e PH states:
"Hit points (hp) measure your ability to stand up to punishment, turn deadly strikes into glancing blows, and stay on your feet throughout a battle. Hit points represent more than physical endurance. They represent your character's skill, luck, and resolve - all the factors that combine to help you stay alive in a combat situation.
So, a high-level 4e character who is low on hit points because he has turned deadly strikes into glancing blows (and is thus covered in nicks and scratches), out of breath from exhausting his physical endurance, and dangerously close to running out of luck is consistent with the 4e definition of hit points.
I like to think of it this way:
Normally, I think I'd do pretty well if I suddenly got into a random fistfight on the street. However, if I go to the gym and lift weights for an hour until my legs and arms feel like jelly, I do not think I would last for more than a few seconds. My reactions are slow, my limbs are weak, and a fresh combatant would probably wipe the floor with me. I figure that that's the point at which I am low on HP. After a short rest I feel much more capable.
Of course, a 4e character is Better Than I Am. He can bend bars/lift gates for hours without tiring. What knocks him out is deflecting the stone-cracking blows of an ogre's hammer or gritting his teeth and covering his face as a
ball cube of white-hot flame licks across his body. At the end of that, he's still standing, but less able to turn away a devastating blow when it comes.
On top of that, for the sake of narrative sense, the devastating blow is almost never automatically lethal. It's a smash to the head that he survives, but perhaps with a severe concussion. A stab to the shoulder that he may or may not bleed to death from. The death save mechanic tells us what sort of wound it is. Die in 3 rounds? You had an artery punctured when you hit 0 HP. Pop back up in 3 rounds after a natural 20? An action-hero style blow to the head that knocked you out, but you shook it off in time to save the day.
Add healing surges to the mix, and it's even better. HP represent the ability to keep fighting. Healing surges represent reserves of will and strength that are depleted over the course of a day. When you wake up, rested and refreshed, you have a full complement of surges. When you take a sharp blow to the head, you might need to take a moment to shake it off (second wind). When your enemy seems to be overwhelming you with a series of almost-lethal blows that you barely manage to deflect, your allies can help you pull yourself together and fight on (e.g. warlord abilities). When you take a splash of acid in the face, it might not do any real damage, but the pain can undermine your ability to fight, something that can be overcome through magically patching up the wound (cleric abilities), distracting you from the pain (warlord abilities), or simply being a badass (second wind).
But there's only so much of this you can pull out of your butt in a day. Eventually, the cleric's magic can't keep up with the latticework of cuts and bruises that covers your body, and the warlord's pep talks start to sound hollow, and you're so tired...so tired...just want to lie down here for a minute until you can feel your hands again...
Healing surges represent real damage accumulating over time. As you become more and more beat up, it eventually becomes impossible to lift your arm to ward off that last blow that takes you down.
Perhaps a character who is out of luck loses his balance and happens to hit his head on a rock, and a character who is out of resolve is simply frightened to death.
I like the 4e system on death and dying for things like psychic damage. What is psychic damage, anyway? Well, in an abstract HP system, it could represent despair, lethargy, involuntary movements and twitches, tunnel vision, or any number of effects that are not physical injury but reduce your ability to keep fighting. If a psychic damage effect puts you below 0 HP, we look to the death save to see what actually happened to you. Did you just faint from the overload (i.e. make your death save), or did you have a stroke and die (i.e fail 3 saves)? It works similarly in 3e, but I enjoy the wildcard that death saves add to the mix.