The dietary properties of minikiwi make them, along with other fruits and vegetables, suitable as the basis for many slimming and pro-health diets. Prolonging the availability of minikiwi can be provided by different storage technologies. This experiment focused on evaluating the effect of various O2 and CO2 concentrations, i.e., low-oxygen atmosphere (DCA, 0.4% CO2:0.4% O2; ULO, 1.5% CO2:1.5% O2) or high-CO2 (CA, 5% CO2:1.5% O2) storage, in order to provide the consumer with fruits with comparable high nutritional values. Evaluation gave the basic characteristics of the fruits that characterize their health-promoting properties, i.e., total polyphenols (TPC), phenolic acids and flavonols, antioxidant activity (AA), monosaccharides, and acid content. The atmosphere with a higher CO2 content of 5% (CA) effectively influenced the high value of ascorbic acid even after 12 weeks of storage. DCA technology contributed to a significant inhibition of phenol loss but not as effectively as CA technology. In contrast, glucose and fructose contents were found to be significantly higher after storage in ULO or DCA, while sucrose content was more stable in fruit stored in CA or DCA. CA technology conditions stabilized the citric acid content of minikiwi, while DCA technology was less effective in inhibiting acid loss. The nutritional value of the fruit after storage in CA or DCA was not significantly reduced, which will allow the supply of fresh minikiwi fruit to be extended and provide a valuable component of the human diet.