We determine the basic physical characteristics of eight gamma-ray bursts (GRBs)—980519, 990123, 990510, 991208, 991216, 000301c, 000926, and 010222—by modeling the broadband emission of their afterglows. We find that the burst kinetic energies after the GRB phase are well clustered around a mean value of 3 × 1050 ergs. In contrast, the energy release in gamma rays, after correcting for a collimated explosion, varies among bursts by more than an order of magnitude. The initial jet apertures are in the 2°-14° range, mildly correlated with the energy, with half of the jets being narrower than ~3°. This implies that, within 100 Mpc, there are about 10 GRB remnants (expanding at ~0.1c) that can be resolved with the Very Long Baseline Array. For all eight afterglows, the total energy in the shock-accelerated electrons is close to the equipartition with protons. However, the slope of the power-law electron distribution is not universal, varying between 1.4 and 2.8. In at least half of the cases, the density structure of the medium is inconsistent with an r-2 profile. A homogeneous medium with density in the 0.1-50 cm-3 range can accommodate the broadband emission of all afterglows, with the exception of 990123, for which we find the density to be less than 10-2 cm-3. If GRBs arise from the core collapse of massive stars, then such low densities indicate the existence of superbubbles created by the supernovae and winds within a cluster of massive stars.