The single-phase titanium ß-alloy Ti10V2Fe3Al (Ti-1023) has been widely used in the aerospace industry due to its unique mechanical properties, which include high fatigue strength and fracture toughness, as well as high corrosion resistance. On the other hand, these unique properties significantly hinder the cutting processes of this material, especially those characterized by a closed machining process area, such as drilling. This paper is devoted to the study of the short hole drilling process of the above-mentioned titanium alloy using direct measurements and numerical modeling. Measurements of the cutting force components in the drilling process and determination of the resultant cutting force and total cutting power were performed. The macro- and microstructure of chips generated during drilling were analyzed, and the dependence of the chip compression ratio and the distance between neighboring segments of serrated chips on cutting speed and drill feed was determined. Experimental studies were supplemented by determining the temperature on the lateral clearance face of the drill’s outer cutting insert in dependence on the cutting modes. For the modeling of the drilling process using the finite element model, the parameters of the triad of component submodels of the numerical model were determined: the machined material model, the model of contact interaction between the tool and the machined material, and the fracture model of the machined material. The determination of these parameters was performed through the DOE sensitivity analysis. The target values for performing this analysis were the total cutting power and the distance between neighboring chip segments. The maximum deviation between the simulated and experimentally determined values of the resulting cutting force is no more than 25%. At the same time, the maximum deviation between the measured values of the temperature on the lateral clearance face of the drill’s outer cutting insert and the corresponding simulated values is 26.1%.