Going to raise the dead real fast.
I have a friend who is well on his way to becoming an Engineer. Part of the degree is designing a product, and he asked me to help. This forum (that I actually first read years ago) got me curious about body armor in general. I'm going to try and keep this as short as possible, while being as informative as I can.
To stop a bullet, you have to do one of two things. Break it, or slow it down REALLY REALLY quickly. Your bones are as brittle as chalk, so we're trying to stop a speeding bullet from breaking grandma's fine china here.
Modern ceramic armor is extremely hard, but also extremely brittle. This is why modern ceramic plates do an awesome job of catching a small number of bullets. The kevlar carriers that hold the plates actually do very little in regard to the bullet, they're there to make sure that the PLATES don't kill you. You see, when the plates shatter, they become needle-sharp shards of ouch. It would suck to stop the bullet, only to have it shove a mass of ceramic knives into you. Once the plates are broken, they cease to do their job. When I was in the army, every ceramic plate had stamped on it "FRAGILE, DO NOT DROP".
Now, you can also buy STEEL armor, which is a specially crafted steel insert. NOT something you could just hammer out in your garage. It's designed to take hit after hit after hit, unlike ceramic variants. Why don't we just issue THAT to the soldiers? Well, they're also considerably heavier. IIRC, ceramic plates are something like 6 lbs, while steel plates can be 16 or more. Quite the difference. Do they protect better? Well, being able to take repeat hits is kind of awesome... but it comes with a slide flaw... it BREAKS the bullet and redirects the pieces. This is called "spall", and can be as dangerous as the bullet itself. You may stop the bullet, only to catch a spray of shrapnel under your chin. This is what "spall guards" are for. Anyway, if your armor is designed to break and deflect bullets, it's worth considering a way to catch spall. Again, kevlar carriers help with this.
Now, both of these approaches break the bullet up. What about slowing it down? "Soft" armor (what this thread is discussing) does this. Kind of like how a soccer goal's net catches and disperses the energy of the ball. Modern rounds are shaped oh so aerodynamically, aren't they? Tapered and slim... how on Earth are we going to defeat that?
The "arapaima" is a fish with unique armored scales that allow it to swim untouched by the local piranha. The scales feature a hard outer layer over a softer interior, allowing the harder outside to "give" a little against the teeth of an attacker. In fact, this "hard on soft" approach lends itself quite well to armor.
Well, first of all, nothing is perfect. The flight pattern of a bullet is spiral, but it is constantly twisting. The impact of a bullet striking something hard causes deformities in the bullet instantly. In physics, the harder object "wins" when two objects collide, regardless of the speed at which the softer object travels. Based on thickness and hardness of the material, the bullet may yet penetrate, and that's to be expected of an object traveling at 13,000 feet per second. However, once it strikes a hard surface, the bullet ceases to be so perfectly shaped and pretty. This irregular shape then becomes very hard to push through additional material, and it gives up energy quite fast.
On an episode of Mythbusters, this was demonstrated. A single phonebook was sufficient to stop up to high calibur rifle rounds, PROVIDED that bullet struck something hard first. The "something hard" in the test? A thin piece of glass from a car window. That's it... GLASS was sufficient to deform the bullet!
That said, if we are created a post-apocalyptic vest for some degree of protection, you'd want a carrier jacket of thick woven material, a steel plate, followed by the sand-caulk-felt backing. All in all, yes, it would be rather thick, hot, and uncomfortable... but you wouldn't wear it all the time anyway... only when you're likely to be on the damaging end of things.