If a particular car doesn't have ABS but there was a version of it in the same model year that had ABS (or close enough to the same model year that "it's the same car"), then generally you can obtain and install all the ABS-related parts (hubs, sensors, wiring harness, controller, probably master cylinder, brake lines, deletion of proportioning valve, add ABS pump and valve module, probably instrument cluster to include the ABS warning lamp). Even this is no small feat because these components are spread out all over the car.
Putting ABS on the above-mentioned 1982 Plymouth Arrow (or any other car that was built with no consideration whatsoever for ever being equipped with ABS) would be a challenge.
As for the original poster, a vehicle that is tail-happy and wants to swap ends without ABS needs attention to the front/rear brake bias, as others mentioned. Even without ABS, it should not be like this. The front wheels should always be arranged to lock first. I've had some production cars (first-gen Honda Civic) that had too much rear brake bias, and it is indeed a handful. If the vehicle in question has switchable ABS (which I've never seen! I wish I could switch ABS off in my car, but I can't) then it's possible that the system is designed so that the ABS is responsible for controlling the rear brake bias. In other words, the rear is over-braked by design, and the ABS is used to keep it under control. This is how so-called "electronic brake force distribution" works - they just chucked out the old fashioned mechanical proportioning valve and let the ABS controller deal with it.
Race cars don't commonly have ABS ... in many cases the rules require that an original-equipment ABS be disabled.
