Hydraulic dampers for the vibration damping of industrial machinery and building structures are typically cylindrical. This study proposes a novel, axially free-folding hydraulic damper of the origami type to improve the structural characteristics of the conventional cylinder shape with restricted effective stroke in relation to the overall length. First, the basic design equation of the proposed origami hydraulic damper was derived by demonstrating that the fold line cylinders on the sidewalls will always meet the foldable condition of the origami hydraulic damper, that is, α=π/n and π/2n≤β≤π/n. Next, the fluid flow characteristics inside the origami hydraulic damper and in the flow path were analyzed; it was determined that the actual damping force exerted on the origami damper was proportional to the square of the velocity of motion. Equations of motion were developed considering the derived damping force equation, and a vibration analysis method using the Range–Kutta numerical analysis technique was established. A validation test system with an origami hydraulic damper in a mass-spring vibration system was developed, and vibration tests were performed with actual seismic waves to verify the damping characteristics and effectiveness of the origami hydraulic damper. Furthermore, the orifice hole diameter at the end of the origami structure as well as the type of internal fluid, were varied in the vibration tests. The effect of the main components of the origami hydraulic damper on the damping effect was analyzed, revealing that the orifice hole diameter had a more significant effect than the internal fluid.