Recently Kuramoto and one of the present authors have carried out a computer simulation for a chemically oscillating system and found a turbulence-like behavior similar to the hydrodynamic turbulence. The steady turbulent state of this system is theoretically studied. It is shown that there exist two characteristic regions of wavenumber k. One is a cascade region with kξ≪1, and the other is a dissipative region with kξ> 1, where ξ is a characteristic length which is much larger than the reaction mean free path lr. Over these two regions the characteristic frequency Ωk and the variance χk of phase fluctuations obey the scaling laws Ωk=k4F(kξ) and χk=k3G(kξ), where F(x) and G(x) are universal functions of x. In a cascade region it is shown that F(x)=c1x-5/2 and G(x)=c2x-5. In a dissipative region with kξ≫1, it is found that G(x)=1 and G(x)=c3 exp (-c4x). An explicit calculation of Ωk and χk over the whole k region is carried out in the Kraichnan direct interaction approximation, and the results are compared with the computer simulation.