(Abridged) In the binary radio pulsar system J0737-3039, the faster pulsar A is eclipsed once per orbit. We construct a simple geometric model which successfully reproduces the eclipse light curves, based on the idea that the radio pulses are attenuated by synchrotron absorption on the closed magnetic field lines of pulsar B. The model explains most of the properties of the eclipse: its asymmetric form, the nearly frequency-independent duration, and the modulation of the brightness of pulsar A at both once and twice the rotation frequency of pulsar B in different parts of the eclipse. This detailed agreement confirms the dipolar structure of the stars poloidal magnetic field. The model makes clear predictions for the degree of linear polarization of the transmitted radiation. The weak frequency dependence of the eclipse duration implies that the absorbing plasma is relativistic, with a density much larger than the corotation charge density. Such hot, dense plasma can be effectively stored in the outer magnetosphere, where cyclotron cooling is slow. The gradual loss of particles inward through the cooling radius is compensated by an upward flux driven by a fluctuating component of the current, and by the pumping of magnetic helicity on the closed field lines. The trapped particles are heated to relativistic energies by the damping of magnetospheric turbulence and, at a slower rate, by the absorption of the radio emission of the companion pulsar.
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