This paper presents a study on the constrained exact boundary controllability of a quasilinear model in fluid mechanics, specifically the Euler equation that describes the motion of inviscid fluids. Our main contribution is the proof of the existence of continuous solutions of the quasilinear model that are Lipschitz continuous with respect to the space variable that take into account lower and upper bounds for the pressure and velocity of the gas flow. These constraints are crucial for the operation of gas pipelines and for ensuring the physical validity of the model. Additionally, we demonstrate the constrained exact boundary controllability of the system under the same pressure and velocity constraints. Our results have potential applications in the control and operation of gas pipelines.