Cancer is still one of the major diseases worldwide. The discovery of new drugs and the improvement of existing ones is one of the areas of priority in the fight against cancer. Dioxadet ([5-[[4,6-bis(aziridin-1-yl)-1,3,5-triazin-2-yl]amino]-2,2-dimethyl-1,3-dioxan-5-yl]methanol) represents one of the promising 1,3,5-triazine derivatives and has cytostatic activity towards ovarian cancer. In this study, we first report the development of dioxadet-bearing nanomedicines based on block-copolymers of poly(ethylene glycol) monomethyl ether (mPEG) and poly(D,L-lactic acid) (PLA)/poly(ε-caprolactone) (PCL) and then conduct an investigation into their characteristics and properties. The preparation of narrow-sized nanoparticles with a hydrodynamic diameter of 100–120 nm was optimized using a nanoprecipitation approach. Thoughtful optimization of the preparation of nanomedicines was carried out through adjustments to the polymer’s molecular weight, the pH of the aqueous medium used for nanoprecipitation, the initial drug amount in respect to the polymer, and polymer concentration in the organic phase. Under optimized conditions, spherical-shaped nanomedicines with a hydrodynamic diameter of up to 230 nm (PDI < 0.2) containing up to 592 ± 22 μg of dioxadet per mg of polymer nanoparticles were prepared. Study of the drug’s release in a model medium revealed the release up to 64% and 46% of the drug after 8 days for mPEG-b-PLA and mPEG-b-PCL, respectively. Deep analysis of the release mechanisms was carried out with the use of a number of mathematical models. The developed nanoparticles were non-toxic towards both normal (CHO-K1) and cancer (A2780 and SK-OV-3) ovarian cells. A cell cycle study revealed lesser toxicity of nanomedicines towards normal cells and increased toxicity towards cancer cells. The IC50 values determined for dioxadet nanoformulations were in the range of 0.47–4.98 μg/mL for cancer cells, which is close to the free drug’s efficacy (2.60–4.14 μg/mL). The highest cytotoxic effect was found for dioxadet loaded to mPEG-b-PCL nanoparticles.