I was bored, so I decided to calculate the greatest mass that you can possibly withstand in an unmodified version of Minecraft.
The most dense material in Minecraft is gold. The density of gold is equal to 19.30 g/cm³. Therefore, the mass of a cubic metre of gold is 19.3 tonnes (metric tons). I will assume that the gold you smelt in Minecraft is in its pure form.
The player's inventory is composed of 36 slots, each slot having a maximum capacity of 64 cubic metres of gold blocks.
Mass(inventory) = 19.3 * 36 * 64 = 44,467.2
44,467.2 tonnes can be carried in the player's inventory.
There is also another four slots for the player's armour, and gold can be used as such.
After careful measurements while using Minecraft's default texture pack, I have come to the conclusion that the thickness of the player's armour is equal to a third of the length of a pixel, or a 48th of a metre (as, proportionally, one square Minecraft metre = 16*16 pixels), so:
Depth = 1 / 48 m.
I also know that the area of the player's armour is composed of 928 pixels (after counting the pixels that we can see in the armour pictures in the minecraft.jar compressed folder). Since the pixel is actually a square unit, I will use metres instead for more clarity, given that the length of a pixel is equal to a 16th of a Minecraft metre.
Conclusion: Steve ≥ Chuck Norris.
I know, this is completely useless information. However, if anybody actually cared and paid attention to any of that, feel free to point out any errors I might have made if you find any.
**Thanks to wells123456 and some1_good for pointing out a couple of careless mistakes I made.
As rickput7 pointed out, the four crafting squares could be filled with gold blocks as well by throwing four stacks on the floor, placing another four stacks in your inventory, then placing them in your crafting slots while picking up the previous four stacks from the floor.
This would add 64 * 4 * 19.3 = 4,940.8 tonnes to the result.
44,468.65755 + 4,940.8 = 49,409.45755 tonnes.
There has also been some speculation on whether or not golden apples are denser than gold as they are crafted with 154.4 tonnes of gold, and the glow which the apple emits would suggest a magical property in said item, so it wouldn't be too far-fetched to think that the apple would weigh its initial mass plus the mass of the gold used to craft it despite its small size.
N-Man has also pointed out that we don't know the density of non-existent fantasy materials such as glowstone and netherrack, so gold being the densest material is only an assumption. However, Crunkatog has made a very logical post concerning his opinion on this matter.
There have been a few complaints from people accusing me of "ripping off" other people's work. To clear out any further confusions, I would like to point out that firstly, I didn't know that this had been done before and I have all the calculations to back up the fact that this work was done from scratch anyway.
Secondly, the results that you would find on the wiki or on other threads are slightly different from mine. You can prove this to yourself by copy/pasting my results + the keyword "Minecraft" in Google, but if you also want to know why they are different, it is simply because the people who have done this before have calculated the mass of each gold ingots used to create the armour while I have calculated the mass of what we can actually see (i.e. gold can't be denser than gold so the result they obtained for the armour's mass, 51.36 tonnes, would be inconsistent compared to the volume of the golden armour).
Lastly, this isn't something to be so worked up over; this was only done for pure self-entertainment (yes, because as I said, I was very bored). Thank you for your understanding.
Also note that carrying 2304 cubic meters of anything in ones pocket would make that person's pocket over many times denser than the core of the sun, therefore compressing something beyond its definable mass and obliterating you and everything around you in a 35 mile radius. How steve manages to create supersaturated solids is beyond me, but I assume it has to do with magnets Hugh Laurie.