In the present study, a thorough examination of nanoindentation-based mechanical properties of a refractory MoTaNbVW high-entropy alloy (RHEA) was conducted. Basic mechanical properties, such as the indentation modulus of elasticity, indentation hardness, and indentation-absorbed elastic energy, were assessed by means of different input testing variables, such as the loading speed and indentation depth. The obtained results were discussed in terms of the elasto-plastic behavior of the affected material by the indentation process and material volume. Detailed analysis of the RHEA alloy’s nanoindentation creep behavior was also assessed. The effect of testing parameters such as preset indentation depth, loading speed, and holding—at the creep stage—time were selected for their impact. The results were explained in terms of the availability of mobile dislocations to accommodate creep deformation. Crucial parameters, such as maximum shear stress developed during testing (τmax), critical volume for dislocation nucleation (Vcr), and creep deformation stress exponent n, were taken into consideration to explain the observed behavior. Additionally, in all cases of mechanical property examination and in order to identify those input testing parameters—in case—that have the most severe effect, an extensive statistical analysis was conducted using four different methods, namely ANOVA, correlation matrix analysis, Random Forest analysis, and Partial Dependence Plots. It was observed that in most of the cases, the statistical treatment of the obtained testing data was in agreement with the microstructural and metallurgical observations and postulates.