No, guests can't bring their bags on board. In fact, there is a metal detector to make sure there's nothing in the guest's pockets or on their person that can fly out and injure another guest.
Before the metal detectors, guests will encounter small 2 way lockers. Put a small bag or item in the locker and when you disembark you will be able to pick it up on the other side.
If you have a small medical bag, or an insulin pump that you want to disconnect and store, you can talk to a staff member and they'll instruct you to talk to a colleague at the boarding area to store the items in a medical locker.