The aim of this study was to produce oleogels based on a vegetable oil blend (high-oleic sunflower, safflower, and linseed oils at a ratio of 60:30:10, respectively) structured with a binary beeswax–monoglyceride system (1:1), as well as to examine the effect of the total structuring agent concentration (3, 6, 9 and 12% (w/w)) on the textural, oil-binding, microstructural, thermal, and rheological characteristics of the resulting oleogels, in order to determine the rational total concentration of the binary structuring agent system. The structuring capacity of the binary system was concentration-dependent: at 6 and 9% (w/w), the hardness of the oleogels exceeded the values of the corresponding single-component systems, whereas at 12% (w/w), no further increase in hardness was observed compared with the monoglyceride oleogel. Increasing the concentration altered crystal morphology: at 6 and 9% (w/w) a more uniform, finely dispersed structure was formed, whereas at 12% (w/w), crystal coarsening and elongation were noted. Overall, a total concentration of 6% (w/w) was considered rational, providing the required balance of characteristics at a relatively low structuring agent content. The resulting oleogel may be regarded as a potential fat base for further research on the development of spreads.