To keep you from falling through the chair, the atoms do not need to "take up space." They merely need to provide a force. In particular, they need to provide exactly enough force to oppose the force of gravity pulling on you.
Fundamental forces like electrostatic forces and gravity don't require "touching" in the intuitive macroscopic sense. They get to convey force at a distance. So they don't need to "fill" the space, as much as they just need to convey enough force.
Where this becomes unintuitive is in the distances involved. The electrostatic forces rise sharply as you get closer to the atoms in the chair (why this happens is another topic entirely). As a result, the difference between a separation that yields a very small force from the chair and a separation that yields a gargantuan force from the chair is very small. We're talking nanometers, or even less. So to our macroscopic-oriented minds, it "feels" like we're making contact, and that contact is hard and unyielding. In reality, the atoms in our body are still a distance away from the atoms in the chair, and there's a little bit of springiness to the interaction. Its just over such small scales that you and I don't intuitively realize it's there.
Its just one of the examples where the intuition that serves us well in daily life just doesn't quite do the job when we start looking at the really small. (The same happens when we start looking at the really large, like planets or stars)