TL;DR It Never Hurts to Ask

The Goblin Is Not Required to Die at 0 HP, even though most do

MM p. 7

A monster usually dies or is destroyed when it drops to 0 hit points.

PHB p. 198

Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws. Mighty villains and special nonplayer characters are common exceptions; the DM might have them fall unconscious and follow the same rules as player characters.

1. For the case where the goblin isn't reduced to 0 HP, this edition isn't that granular in combat mechanics. Any creature with > 0 HP functions fully until that 0 HP mark is reached. (See below for asking anyway, and a good use of inspiration if that's being used at your table). Since HPs are an abstraction, note that PCs by these rules don't get weaker the closer they get to 0 HP. Most rulings would be "the goblin's hit, and he keeps running."

2. At 0 HP: the goblin isn't required to die at 0 HP. In a given situation, this monster/NPC could be treated as a PC is, but the player needs to make clear to the DM what he's trying to do. If the DM agrees/rules favorably, there are a variety of ways to keep the goblin alive once you bring him down. Depending upon your party make up, you can attempt to

   • use a healing spell,
   • use a healer kit,
   • cast the Spare the Dying cantrip,
   • a healing potion,
   • attempt to stabilize the goblin (PHB p.197).

   Depending on the methods available and attempted, the DM may need to roll a death save or two on the goblin's behalf to see if these attempts are successful.

3. In a comment in this answer to this question @anaximander used this reasoning:

   The main reason monsters just die at 0HP is because it's more to keep track of and is almost always unnecessary. There is a reason for the goblin to not die immediately, so the death save / Spare The Dying mechanic has a point here. Declaring "I want to interrogate, or revive, that goblin" makes that enemy "special".

   Declaring "I want to stabilize that goblin and keep him from dying" after bringing him down with the bow shot is a valid player attempt. It has rules support in this edition. The "called shot" doesn't have explicit rules support in this edition, but the DM can still support a called shot.

4. The DM can rule in favor of a called shot as you described it. It never hurts to ask. DM will likely either assign disadvantage or make other modifications to the attempt since it's a harder shot than the regular attack with a ranged weapon. He could have you use a DM assigned inspiration point that was previously earned (DMG p. 240-241) for the attempt. He could apply the "rule of cool."

5. Is a DM required to allow any of that? No, but the basics of the game provide a solid premise for this to work. (It never hurts to try!). I'll predict that most DM's would support the attempt, as it fits very well into the fundamentals of how the game is played: (PHB p. 6)

   • The DM Describes the Environment: "Your arrow hit the Goblin and he goes down."

   • The Players describe what they want to do: "(Ranger) I want to run to the goblin and stabilize him so that he doesn't die. We need a prisoner!" (Cleric) "I intend to use my Spare the Dying cantrip on the goblin!"
     - The DM narrates the results of the adventurers' actions:

     • (rolls one death save) "You arrive at the goblin and he's still, not breathing. When the cleric's "Spare the Dying" cantrip goes off, you notice him begin to breath again."
     • "You arrive at the goblin, and he's still. Your arrow entered in his back and came out where a goblin's heart is. Your attempts to revive him do not succeed." (Possible follow up is cleric using a healing spell, with another ruling by the DM. Success or failure?)
     • "You arrive at the goblin, and he's still. Your arrow entered in his back and came out the front. Your efforts to stabilize him seem to be working, as he begins to twitch and moan when you apply the combat first aid you learned in ranger school."
     • some other result

6. An impromptu called shot would go similarly.

   • The DM Describes the Environment: "The Goblin is fleeing at top speed!"
   • Player describes action "Using my long bow, I shoot to wound/disable. Trying to hit him in the leg/knee." (Possibly adding "Going to burn that inspiration point I earned earlier.")
   • The DM narrates the results of the adventurers' actions: "Roll your attack with disadvantage." or "Using that inspiration point, make the attack to kneecap him with your usual attack, as it overcomes disadvantage ..." (Ranger rolls, results in hit or miss, and the game proceeds ...

While you specified a ranged attack, if your Ranger can Misty Step or otherwise close to within melee range, the melee option to knock out would apply. (PHB p. 198.)

When an attacker reduces a creature to 0 hit points with a melee attack, the attacker can knock the creature out. The attacker can make this choice the instant the damage is dealt. The creature falls unconscious and is stable.