Is it possible to do a barrel roll in a large aircraft like a B737?
Yes, it has been done before in an even larger aircraft, the Boeing 367-80, very famously by "Tex" Johnston on August 7, 1955. This aircraft was the prototype for the Model 707. 

When a barrel roll maneuver is properly performed the aircraft remains in a positive-G state, so no "inverted flight" is experienced by the aircraft. According to the International Aerobatic Club,


  The Barrel roll is a combination between a loop and a roll. You complete one loop while completing one roll at the same time. The flight path during a barrel roll has the shape of a horizontal cork screw. Imagine a big barrel, with the airplanes wheels rolling along the inside of the barrel in a cork screw path. During a barrel roll, the pilot always experiences positive Gs. The maximum is about 2.5 to 3 G. The minimum about 0.5 G.