We have studied the structure of chaos in three-body dynamics using the concept of intermittency, implying that violent states of a system alternate in time with quasi-regular states producing together a non-stationary and evolving pattern of unpredictable behaviour. Computer simulations are produced to demonstrate explicitly sporadic short violent bursts in quasi-regular hierarchical states of the systems. This is seen both in orbits and in the long time series generated by the system. The time series prove to be similar in shape to what is observed in various physical experiments with laboratory chaotic systems when they reveal the so-called type-III intermittency. The new effective methods of time series analysis enable us to discover a strange attractor with a fractal dimension slightly above 2. This shows that three-body dynamics has the same intrinsic qualitative structure and quantitative measure of chaos as the widely known chaotic system, the Lorenz attractor.