In external flow simulations, the selection of domain size is crucial for achieving accurate results while managing computational efficiency. This study investigates the effects of domain size on simulation outcomes, highlighting the challenges associated with quantifying these effects. A narrow domain can significantly distort simulation results, whereas an excessively large domain can lead to increased computational costs. This duality necessitates a careful balance, as quantifying the domain effect is essential yet challenging due to the difficulty in isolating it from other influences, particularly mesh effects. Without an objective reference, assessing the impact of domain size becomes problematic. To address these issues, we first explore the effects of domain size through two manufactured problems and a benchmark case involving low-speed flow past a cylinder. Our findings confirm that the use of finite domains introduces errors, which can only be mitigated by enlarging the domain. However, it is important to note that increasing the domain size without refining the mesh yields limited benefits, as the existing mesh may not adequately resolve additional flow features. This highlights the importance of mesh refinement in conjunction with domain size adjustments to enhance simulation accuracy. Furthermore, we employ a physics-informed neural network (PINN) approach to quantify the domain size effect. By utilizing a parameterized governing equation that models an infinite domain within a finite space, the PINN framework effectively incorporates the domain effect into its numerical error estimation. This innovative methodology allows for a more nuanced understanding of how domain size influences overall simulation accuracy. Through mathematical derivation, we isolate domain-specific errors from other types of numerical errors, providing a clearer picture of the contributions of domain size to simulation inaccuracies. Our results demonstrate that the PINN approach consistently and accurately estimates the effects of domain size, both as an integral part of the overall numerical error and as an independent contribution. This estimation remains robust across various domain and mesh sizes, with finer meshes resulting in tighter error bounds. The ability of PINNs to incorporate complex domain effects into their error estimation presents a significant advancement in the field of computational fluid dynamics.