"I'm not doing anything" would be better. You would be understood by saying "I'm doing nothing" but it would sound a little strange. Actually, most native English speakers would probably add "at the moment". So the best answer would be "I'm not doing anything at the moment." The question "What do you do?" would usually be interpreted as meaning "What work do you do?"

Nothing in this context is a NOUN. For example, A: What are you doing? B: I am washing. << Washing becomes the name of the activity being done>> Thus in this case, nothing "is a noun, the name of a state of inactivity" So technically, It is correct to say both. ^..^

Both answers are correct .