Using an artificial neural network (ANN), the values of the antiradical potential of 1315 items of food and agricultural raw materials were calculated. We used an ANN with the structure of a “multilayer perceptron” (MLP) and with the hyberbolic tangent (Tanh) as an activation function. Values reported in the United States Food and Nutrient Database for Dietary Studies (FNDDS) were taken as input to the analysis. When training the ANN, 60 parameters were used, such as the content of plastic substances, food calories, the amount of mineral components, vitamins, the composition of fatty acids and additional substances presented in this database. The analysis revealed correlations, namely, a direct relationship between the value of the antiradical potential (ARP) of food and the concentration of dietary fiber (r = 0.539) and a negative correlation between the value of ARP and the total calorie content of food (r = −0.432) at a significance level of p < 0.001 for both values. The average ARP value for 10 product groups within the 95% CI (confidence interval) was ≈23–28 equivalents (in terms of ascorbic acid) per 1 g of dry matter. The study also evaluated the range of average values of the daily recommended intake of food components (according to Food and Agriculture Organization—FAO, World Health Organization—WHO, Russia and the USA), which within the 95% CI, amounted to 23.41–28.98 equivalents per 1 g of dry weight. Based on the results of the study, it was found that the predicted ARP values depend not only on the type of raw materials and the method of their processing, but also on a number of other environmental and technological factors that make it difficult to obtain accurate values.