[Abridged] We present an 8.5-hour simultaneous radio, X-ray, UV, and optical observation of the L dwarf binary 2MASSW J0746+20. We detect strong radio emission, dominated by short-duration periodic pulses at 4.86 GHz with P=124.32+/-0.11 min. The stability of the pulse profiles and arrival times demonstrates that they are due to the rotational modulation of a B~1.7 kG magnetic field. A quiescent non-variable component is also detected, likely due to emission from a uniform large-scale field. The H-alpha emission exhibits identical periodicity, but unlike the radio pulses it varies sinusoidally and is offset by exactly 1/4 of a phase. The sinusoidal variations require chromospheric emission from a large-scale field structure, with the radio pulses likely emanating from the magnetic poles. While both light curves can be explained by a rotating mis-aligned magnetic field, the 1/4 phase lag rules out a symmetric dipole topology since it would result in a phase lag of 1/2 (poloidal field) or zero (toroidal field). We therefore conclude that either (i) the field is dominated by a quadrupole configuration, which can naturally explain the 1/4 phase lag; or (ii) the H-alpha and/or radio emission regions are not trivially aligned with the field. Regardless of the field topology, we use the measured period along with the known rotation velocity (vsini=27 km/s), and the binary orbital inclination (i=142 deg), to derive a radius for the primary star of 0.078+/-0.010 R_sun. This is the first measurement of the radius of an L dwarf, and along with a mass of 0.085+/-0.010 M_sun it provides a constraint on the mass-radius relation below 0.1 M_sun. We find that the radius is about 30% smaller than expected from theoretical models, even for an age of a few Gyr.
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