Background: The production of reactive oxygen species (ROS) in animals and cells often results from exposure to low-intensity factors, including magnetic fields. Much of the discussion about the initiation of oxidative stress and the role of ROS and radicals in the effects of magnetic fields has centered on radical-induced DNA damage. Methods: The DNA concentration in the final solution was determined spectrophotometrically. Typing of the polymorphic variant rs1052133 of the 8-oxoguanin DNA glycosylase (hOGG1) gene was performed by polymerase chain reaction. An enzyme immunoassay was performed to determine the level of 8-oxyguanine in DNA. To process samples exposed to an alternating magnetic field, the authors developed a device for the automated study of biological fluids in an alternating magnetic field. The content of hydrogen peroxide in aqueous solutions of DNA was determined using the spectrophotometric method. Results: It was experimentally determined that an increase in the concentration of hydrogen peroxide in an aqueous medium by 3–5 times under the action of a low-frequency magnetic field reduces the resistance of the genomic material to oxidative modification and the accumulation of 8-oxyguanine in DNA. A model is proposed for the mechanism of action of a low-frequency magnetic field on aqueous solutions of nucleic acids and proteins, which satisfies the model of a chemical oscillator for the transformations of reactive oxygen species in an aqueous medium. The model illustrates the oscillating nature of the processes occurring in an aqueous solution of DNA and makes it possible to predict changes in the concentration of hydrogen peroxide in an aqueous solution of biopolymers, depending on the frequency of the acting low-intensity magnetic field. Conclusions: The key element in the mechanisms involved in the effects of low-intensity magnetic field on living systems is the occurrence of ROS generation in the aquatic environment of chemical oscillators, in which the competition of physical and chemical processes (electron transfers, reactions of decay and addition of radicals, spin magnetically induced conversion, synthesis, and decay of the longest-lived form—hydrogen peroxide) is controlled by a magnetic field.