This is an interesting question. You're correct on your estimate of the weight of 36 cubic feet of water at about 2200 pounds, and if you were to lay down and place a hard bottomed
tank of water this size on your chest you would feel the weight of all of it.
If however, the bottom of the tank was soft and pliable, more like the surface of a water balloon, you would feel significantly less weight as it forms around you. I'd estimate between a quarter to half of the area would be supported by the ground around you, however 1100 pounds does still sound like quite a lot. The factor you really need to consider in this case is pressure. With a soft bottom like that I mentioned all that force would be evenly spread out over every square inch of your body, sticking with the estimate of about 6 square feet, or 864 square inches, that means you only feel an extra pressure of 1100/864=1.27 PSI at a depth of 3 feet.
This soft bottomed tank can be considered an analogy to a person laying at the very bottom of a swimming pool, with their back against the floor. But if there is any water in between your body and the floor at all then we need to consider a piece of fluid mechanics which basically can be summed up as "Pressure in a fluid is exerted in all directions evenly." This means with water completely surrounding you, you don't really feel any net force on your body from the weight of the water like you would with the whole tank sitting on your chest because the water is pushing as much up as it is down, and indeed from all sides.
(In fact, you are currently in a large pool of air. It's weight exerts a pressure of about 15 PSI on you from every direction. You're probably used to it./mbw)
You do feel this pressure though, if you go swimming and dive you will likely be able to feel it on your eardrums pretty quickly, and if you go deep enough you can feel your body actually being compressed. For a dramatic example imagine tying a balloon to a cinderblock and throwing it in a lake, as it sank the balloon would actually shrink until the water pressure was equalized by the internal pressure from the air inside. The same thing happens to our bodies but it's a bit harder to notice.
(published on 02/22/2012)