A dragonfly flies because of the movements that its wings make. This is entirely different to a manmade aircraft powered by a jet engine which effectively uses reaction thrusters to provide propulsion. However, it is possible to have a "jet" powered aircraft, such as many helicopters that use jet turbines as a source of mechanical power.
However, if we were to scale up a mechanical dragonfly to a size where it could carry a human, we would encounter problems related to Reynolds Number. The effect of Reynolds Number on wings is that for insect-sized airfoils, they may be thin and irregular in cross-section, while human-scale aircraft wings must be thick and smoothly curved. This in itself adds significant weight to the wings, not considering the square-cube law that says that larger objects must have stronger, heavier internal structures to support themselves.
Dragonflies rely upon flexible connections between their wings and bodies to connect the two, however when scaled up to aircraft size, these mechanical linkages would need to be vastly stronger, and allowing the necessary range of controlled movement would make them quite complex.
Flying creatures fly with flapping wings because of the biological rule that rotational structures on any level above the sub-cellular are not practical. This causes inefficiencies to do with the change in the momentum of the wings twice per stroke cycle. To scale up beating wings would only magnify this problem. However, man-made mechanical systems do not have such limitations, and rotating systems have the advantage of conservation of momentum.
So, while it might be possible to build an aircraft-scale jet-turbine powered four-winged ornithopter, it would be mechanically complex, difficult to achieve controlled flight (requiring computerisation to convert control input into wing movements), very energy inefficient (giving it an abysmally short operational duration) and highly prone to wear and catastrophic mechanical failure at the joints between the wings and fuselage, and they would be entirely unnecessary when we already have aircraft that perform in a similar manner: Helicopters.