Dudaim melon (Cucumis melo Group Dudaim) is a unique edible melon for which few postharvest physiology studies have been conducted. To investigate the postharvest behavior of dudaim melon, two cultivars (Zangi-Abad and Kermanshah) were planted, tagged at anthesis, and harvested at two maturity stages: 21 and 28 d after anthesis (DAA). Harvested fruit were stored at 5 or 13 °C for up to 3 weeks and various quality parameters including color, firmness, titratable acidity (TA), total soluble solids (TSS), weight loss, chilling injury (CI), ethylene production, protein content, glucose content, fructose content, sucrose content, and maltose content were assessed during storage. After 3 weeks of storage at 13 °C, early-harvested fruit (21 DAA) had relatively similar color values (L*, lightness; a*, green–red tones; b*, blue–yellow tones) and TA compared with late-harvested fruit (28 DAA); however, some quality traits, such as TSS, were not similar. Ethylene content decreased initially after harvest and then started to increase during storage at 13 °C. For most treatments, glucose and fructose contents decreased whereas sucrose and maltose contents increased with advancing maturity. Increased ethylene production, in concert with color development at 13 °C, similar to ripe fruit, and the changing balance of measured mono- and disaccharide sugars in harvested fruit likely indicates ‘Kermanshah’ is climacteric. Results for ‘Zangi-Abad’ were not as definitive. Dudaim melon fruit can be harvested at an optimum stage of maturity, similar to known climacteric melon fruit, and then allowed to ripen at proper storage temperatures before consumption. Based on the results of this study, we recommend that harvest at 21 DAA and storage at a nonchilling temperature such as 13 °C are the optimal stage and temperature for long storage purposes.