It is possible but not easy or cheap. If you want an heavy bomb shelter then you need lots of concrete walls supporting the interior. It will be like a dungeon.
Contact some commercial contractors where upper stories have concrete floors poured all the time. You have to have special metal trusses with metal sheeting to do it. Contact an architect who does commercial stuff also.
Why not just settle for one small room in the area of the basement for your tornado/bomb shelter. Maybe an room extending out under the porch or patio so the concrete performs a double use.