What do we do with a "stray" cat?

This cat wandered into our house about an hour ago. He's slim, but not bony exactly, and his fur is normal. He doesn't show any obvious signs of maltreatment, so we began to carry him out and place him in the street. Every time he would bolt back in before we could. He was scrabbling at the bin and pulling out anything he could find, and eating anything he could find on the floor. Eventually we decided to feed him some scraps, but we placed them outside along the street so he wouldn't 'adopt' us. He then grabbed a mouthful before retreating inside to our house to eat it in a corner, before going back out to eat more. We don't know where he lives, as he's only wearing a battered collar with a bell on it. WE don't know what to do: we can't keep him overnight, in case he adopts us, and we can't shut him out because he keeps sprinting back in and scratching at the door. It's going to be very, very cold tonight. What do we do?