I hope that the father of the "modern Stone Age family" has thick skin, or else he is going to lose his legs.
Let's put aside the fact that Fred Flintstone basically runs to work and therefore doesn't really need his wheels (or that he would need the quads of a god to get them moving). What is much more interesting is the way he stops his caveman car. With heels screeching and smoking, Fred famously uses his own feet to stop his forward momentum. Much like how your car's brake pads work, Fred's feet absorb all the frictional forces until the stones stop rolling.
But just how much force would his feet need to apply, and what would happen to them?
Friction's Victory Over DaFeet
The coefficient of friction--a ratio of the force friction provides and the weight pressing down on a surface--is the first thing we would have to find. It has been studied for human skin [PDF], but what we are really looking for is the roughness of the road; how much resistance Fred's feet would meet. I doubt that The Flintstones had the paved roads of today, so a first guess at the roughness of the road Fred is braking on would be a bit higher than the values we have on highways: around 0.7-0.9.
Heavy loads encounter more friction, so next we have to estimate the weight of Fred's famous rockmobile (replicas are for sale, by the way).
Most of the car's mass is going to be bound up in the huge rock rollers at the front and back. Let's assume that they are granite. If they were one and a half meters long and 80 centimeters in diameter--like huge stone rolling pins--they would be about 360 kilograms (~795 pounds) each. But that's not all pressing down on the road Fred has to stop on. Fred himself is a hefty fellow, maybe 95 kilograms (210 pounds). For the sake of estimation, we could assume the rest of the car, made of tarp and wood, weighs and additional 50 kilograms (110 pounds). So all in all Fred must use his heels to stop an 865-kilogram (1910 pound) rockmobile. (Note that this is much lighter than cars today due to their metal frames, engines, and other components.)
So we have the weight, now we need the speed. Fred moves his wheels under his own power, but I wouldn't guess he could go too fast. The fastest human powered bicycle has been clocked at an astonishing 82 miles per hour. There is no way a chubby caveman could get a hunk of granite going that fast with only his legs, and even average highway speeds seem out of the question. The rockmobile is neither a picture of aerodynamics or engineering genius, but it still has to be faster than simply running to make any sense at all as a vehicle. I'll give Fred the benefit of the doubt and say his top speed is 25 miles per hour.
Now that we have all the numbers, all we have to do is plug them in. Assuming for a moment that Fred's feet are so callused as to be indestructible, slamming his feet into the rocky road at 25 miles per hour would bring the rockmobile to a stop in about 26 feet (8 meters). That's not too bad considering his bunions are brake pads. (If you want to see the full calculation for stopping distance, I did the same for Batman and the Joker sliding on the floor here.)
But what if Fred's feet weren't indestructible? The force of friction on his feet would be twice the bite force of a large American alligator, so I don't think his toes would take it too well. To put it another way, it would be like standing on a belt sander, with an adolescent African elephant on your back.
The best-case scenario for Fred's feet is that they incur a mild "road rash"--the kind of abrasions and burns that motorcyclists and bikers get when they skid along the ground at speed.
The worst-case is much more likely. If Fred pressed and held his feet to the ground, he would most likely lose them. Sadly, we know this would happen because of the unfortunate cases where people's feet are run over by cars (study link includes some graphic images).
When a material can't handle the friction, the result is catastrophic heating and disintegration. In all likelihood, Fred Flintstone's feet after braking would look like airplane landing gear when it can't handle the friction:
If Fred decides to take his whole family out for a brontosaurus burger and a drive-in movie, the problem of feet-destroying friction only gets worse because the rockmobile would be significantly heavier. All this is assuming of course that the car itself would hold together for more than a few feet. As an engineer, I have no idea how a forward-moving car keeps a rear wheel on that has no backstop.
Engineers today have developed brakes that can take huge amounts of heat and stress precisely because other materials would fail under such strenuous conditions. On a list of those unusable materials is certainly human skin.
Using your bare feet as brakes is a yabba dabba don't.
Images: Screenshot from Flintstones opening credits; Airplane landing gear from Airlinesafety.com