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A variety of current models for gamma-ray bursts (GRBs) suggest a common engine - a black hole of several solar masses accreting matter from a disk at a rate 0.01 to 10 solar masses per second. Using a numerical model for relativistic disk accretion, we have studied steady-state accretion at these high rates. Inside a radius ~ 10**8 cm, for accretion rates greater than about 0.01 solar masses per second, a global state of balanced power comes to exist between neutrino losses, chiefly pair capture on nucleons, and dissipation. Energy emitted in neutrinos is less, and in the case of low accretion rates, very much less, than the maximum efficiency factor for black hole accretion (0.057 for no rotation; 0.42 for extreme Kerr rotation) times Mdot c**2. The efficiency for producing a pair fireball along the rotational axis by neutrino annihilation is calculated and found to be highly variable and very sensitive to the accretion rate. For some of the higher accretion rates studied, it can be several per cent or more; for accretion rates less than 0.05 solar masses per second, it is essentially zero. The efficiency of the Blandford-Znajek mechanism in extracting rotational energy from the black hole is also estimated. In light of these results, the viability of various gamma-ray burst models is discussed and the sensitivity of the results to disk viscosity, black hole rotation rate, and black hole mass explored. A diverse range of GRB energies seems unavoidable and neutrino annihilation in hyper-accreting black hole systems can explain bursts up to 10**52 erg. Larger energies may be inferred for beaming systems.