Background/Objectives: Current promising treatments for many diseases are based on the use of therapeutic nucleic acids, including DNA. However, the list of nanocarriers is limited due to their low biocompatibility, high cost, and toxicity. The design of synthetic building blocks for creating universal delivery systems for genetic material is an unsolved problem. In this work, we propose PAMAM dendrimers with rigid thiacalixarene core in various conformations, i.e., PAMAM-calix-dendrimers, as a platform for a supramolecular universal constructor for nanomedicine. Results: Third generation PAMAM dendrimers with a macrocyclic core in three conformations (cone, partial cone, and 1,3-alternate) were synthesized for the first time. The obtained dendrimers were capable of binding and compacting calf thymus DNA, whereby the binding efficiency improved with increasing generation, while the influence of the macrocyclic core was reduced. A dramatic effect of the macrocyclic core conformation on the hemolytic activity of PAMAM-calix-dendrimers was observed. Specifically, a notable reduction in hemotoxicity was associated with a decrease in compound amphiphilicity. Conclusions: We hope the results will help reduce financial and labor costs in developing new drug delivery systems based on dendrimers.