First thing--the disclaimer--If you are not comfortable fixing brakes yourself, take it to someone who is, as they are the car's primary safety system.

Now, what is the problem. Without looking at the car, it sounds like the problem may not be the light/BAS, but that is a result of the problem. Has the brake fluid ever been changed? Since the car sits A LOT, it may be bad, and may have in turn degraded the brake lines (rubber, not metalic). As the brake hoses age, they get spongy, and constrict (just like human veins before a heart attack. In addition, corrosion can develop in the caliper (especially from old fluid). All this prevents the fluid from returning to the master cylinder after application of the brakes. Pressing the pedal creates enough force to squeeze the caliper, but the caliper does not have enough force to return the fluid.

However, it can also be a problem with the BAS/ABS control block, again, as a result of siting with old fluid. Corrosion. While low miles are good, a car that just sits begins to break down. Cars are very similar to people, no? If the brake fluid was never changed, you need to get the truck somewhere to have that done, and have the hoses and calipers inspected. The system is crudded up.