Minecraft, without the death?

Here's a suggestion:

If you're playing Vanilla, you can always create a set of command blocks that constantly heals and gives resistance to the player.

Firstly, you'll have to start the world in Creative Mode and cheats (you can switch back to Survival once you've complete this setup and play from there; that's one of the reasons you need cheats enabled).

Then, give yourself a command block.
/give @p minecraft:command_block 1

Now, create a loop of command blocks; a clock chain, preferably looping every 3 seconds.

Create command blocks, with the following commands:
/effect @a 10 2401 1 true (Regeneration)
/effect @a 11 2401 100 true (Resistance)
/effect @a 12 2401 1 true (Fire Resistance)
/effect @a 6 2401 20 true (Instant Health)
/effect @a 23 2401 20 true (Hunger Saturation - If you do not want hunger to deplete)
/gamerule keepInventory true (Keeps the inventory and XP on death)

Now, make sure your command blocks do not get destroyed (ie. Protect it by covering the whole contraption with bedrock) and now, you can turn back into Survival mode with /gamemode @p 0 and play on from there.

EDIT: Resistance does protect you from fall damage. It prevents pretty much any form of damage, as long as the protection is high enough.
So, falling from 255 height to 1 can be protected with: /effect @a 11 5 251 (Resistance to 254-3; 125.5 hearts of damage)

As for mods that does this; there's none that I've heard of.

If you would like screenshots or a video tutorial, I don't mind making one.

As of MC 1.8 Potion effects via the effects command is no longer using the ID system. Here are the commands for MC 1.8

/effect @a minecraft:regeneration 1000000 255 true
/effect @a minecraft:resistance 1000000 255 true
/effect @a minecraft:fire_resistance 1000000 255 true
/effect @a minecraft:instant_health 1000000 255 true
/effect @a minecraft:saturation 1000000 255 true

As of MC 1.13, they changed the syntax:

/effect give @a minecraft:regeneration 1000000 255 true
/effect give @a minecraft:resistance 1000000 255 true
/effect give @a minecraft:fire_resistance 1000000 255 true
/effect give @a minecraft:instant_health 1000000 255 true
/effect give @a minecraft:saturation 1000000 255 true

There is a mod called Extra Utilities that, among other things, adds a Peaceful Table. This table allows players who play the game on Peaceful difficulty to get mob drops.

Taken from the wiki directly:

The Peaceful Table is a block added by Extra Utilities, that allows players who play on peaceful to get mob drops. In order for the table to work, it has to be placed in the world adjacent to a chest (or any block with an inventory) which has a sword in it. The table will occasionally spawn a mob and kill it instantly with the provided sword. The sword will take damage appropriate to the hits needed to kill the mob with the sword in the inventory. In other words, better swords last longer. Enchantments are also taken into account. Looting will give extra mob drops. Sharpness and Unbreaking will make the swords last longer.

A Peaceful Table may be placed in the Nether in order to get Nether mob drops; but must be placed in a Nether Fortress in order to get Blaze and Wither Skeleton drops. A Peaceful Table may be placed in the End in order to get Enderman drops.

If you use this mod, craft this table, and have your difficulty set to Peaceful, I see all of your objectives as met.

There is a mod called Gravestone, it adds in a gravestone that will spawn were you die, containing all of your items, indefinitely, or until you come and break it, getting your items back.

This isn't exactly what you asked for, but I suggest that an option worth considering because it is simple and already available in the game is:

/gamerule keepInventory true

Like all gamerules, the effect of this command lasts forever (so no gimmicks with command blocks are needed). You do need to enable cheats to be able to set it (either at world creation time or by temporarily using "Open to LAN").

What it does is, in the event that you die, when you respawn you will still have all of your inventory items and levels. Thus death merely means you teleport back to your last bed or spawn point — since you don't lose any items, you haven't really lost any of your progress in the game. Death is no more inconvenient than falling down a ravine and having to climb your way back up, or getting completely lost in a cave.

If you want to stop the negatives of death try an inventory keeping mod, and a teleporter mod, if you die,you can just respawn at your bed/spawn and hop over to where you were, no progress lost, almost as if you never died.

Becoming invincible is a bad idea for anyone asking, not only does it make mobs laughable (which really does take the "surviving" aspect out of survival mode) , it can also break the game, i mean what happens if you fall into the void?, you'll be stuck.

If you hate lava, Pack some enchanted golden apples using cheats, they'll give you fire resistance long enough. (if you don't like the damage resistance or the regen, you can always pack a stack of spider eyes to reverse the effect).