Contactless optical electroreflectance measurements at different temperatures are used to study exciton states in a structure involving a periodic system of 36 GaAs quantum wells separated by tunneling-nontransparent AlGaAs barriers with thickness 104 nm. In the structure, the width of 32 of the quantum wells is 15 nm, while the width of the remaining four quantum wells, numbered 5, 14, 23, and 32, is 20 nm. The periodicity of the structure corresponds to the Bragg interference condition at the excitonic frequency in quantum wells at the angle of incidence of light ∼43°. From the quantitative analysis of the shape of the contactless electroreflectance line, the parameters of the exciton ground states and excited states are determined for both types of quantum wells. It is established that, for the system of four 20-nm-wide quantum wells separated by a distance of 830 nm, the size-quantization energy in the ground state is 8.4 ± 0.1 meV, and the parameter of broadening of the excitonic peak is 1.8 ± 0.1 meV at 17 K and increases with temperature up to 2.0 ± 0.1 meV at 80 K. For the system of 32 wells with the width 15 nm, the quantum confinement energy in the ground state is 14.9 ± 0.1 meV, and the parameter of broadening of the excitonic peak is 2.2 ± 0.1 and 2.6 ± 0.1 meV at 17 and 80 K, respectively. The possible causes of radiative and nonradiative broadening of exciton states in the systems are discussed.