Translation:My hands and feet both hurt a lot.
Agreed. I would consider the "hands and feet are both painful" to be just plain wrong, on many levels. It's awkward English, and no native speaker would ever say it. Also, "painful" means causing pain. Perhaps Bruce Lee would say that about his hands and feet to warn his enemies? Even then, "both" makes no sense here.
What I don't understand is where the "a lot" comes from here. There are great inconsistencies in how 很 is used in this course. Sometimes in a sentence like the answer above it means "hurt a lot". Other times it just means "hurt".
That makes it very open to any interpretation of degree of hurt or anything else.
疼死, téng sǐ actually means to 'really hurt' and closer to 'hurts a lot'.
生疼 shēng téng means extremely painful.
How about "I hurt in BOTH my hands and feet (arms and legs?)" or "I have limb pain" or "My limbs hurt"?
"My arms and legs both hurt." REJECTED
"My limbs all hurt." REJECTED
"All my limbs hurt." REJECTED
"Both pairs of limbs hurt." REJECTED
"Limbs on both sides hurt." REJECTED
"My hands and feet both hurt." ACCEPTED
For some unknown reason, I want to use "both" even though I have FOUR limbs.
I believe 很 is grammatically required in this sentence. It doesn't necessarily mean "very" here. In most Chinese sentence structures, an adjective-predicate can't stand alone. It must have an adverb with it. 很 is often used just because something has to be put there. It doesn't always mean something as strong as the English "very."