Accessibility of synthetic oligonucleotides and the success of DNA nanotechnology open a possibility to use DNA nanostructures for building sophisticated enzyme-like catalytic centers. Here we used a double DNA crossover (DX) tile nanostructure to enhance the rate, the yield, and the specificity of 5'-5' ligation of two oligonucleotides with arbitrary sequences. The ligation product was isolated via a simple procedure. The same strategy was applied for the synthesis of 3'-3' linked oligonucleotides, thus introducing a synthetic route to DNA and RNA with a switched orientation that is affordable by a low-resource laboratory. To emphasize the utility of the ligation products, we synthesized a circular structure formed from intramolecular complementarity that we named "an impossible DNA wheel" since it cannot be built from regular DNA strands by enzymatic reactions. Therefore, DX-tile nanostructures can open a route to producing useful chemical products that are unattainable via enzymatic synthesis. This is the first example of the use of DNA nanostructures as a catalyst. This study advocates for further exploration of DNA nanotechnology for building enzyme-like reactive systems.