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(Abridged) We describe the results of the highest-resolution radioobservations yet made of four relic radio sources in the Abell clusters A13,A85, A133 and A4038. Our VLA images at 1.4 GHz with 4" resolution show aremarkable variety of fine structure in the form of spectacular arcs, wisps,plumes and loops. Their integrated radio flux densities fall very rapidly withfrequency, with power-law slopes between 2.1 and 4.4 near 1.4 GHz The relicspossess linear polarization levels ranging between 2.3 % (A133) and 35 % (A85);the higher polarization fractions imply a highly ordered magnetic field in thefine structure and low differential Faraday rotation in the intervening clustergas. The optical identification of host galaxies remains problematic. In A85,A133 and A4038 the travel times for the brightest cluster galaxies aresignificantly longer than the modeled ages of the relics and nearby brightellipticals provide a better match. Excess X-ray emission in the 0.5 keV-to-2keV band was found near the relics in A85 and A133. The surface brightness wastoo high to be attributed to the inverse-Compton mechanism alone. We foundexcellent fits to the broad-band radio spectra using the anisotropic (KGKP)model of spectral ageing, and we have extended the model to include diffusionof particles between regions of different field strength (the Murgia-JP, orMJP, model). The steep radio spectra imply ages for the relics of ~ 10^8 yr, atthe start of which period their radio luminosities would have been ~ 10^25 W/Hzat 1.4 GHz.Comment: 43 pages, 13 figures, AJ, Sep 2001 (accepted

Four Extreme Relic Radio Sources in Clusters of Galaxies