Papaya is a tropical fruit that is highly appreciated worldwide for its organoleptic properties and its nutritional and medicinal value. The optimal fruit quality for market requirements is determined by genotype, environmental conditions, and crop management, as well as fruit handling at harvest and adequate postharvest conservation. The aim of this work was to determine the optimal maturity stage, based on skin color, at which to harvest papayas grown in Mediterranean greenhouses with active climate control in south-east Spain, depending on the harvest season. We confirm that it is possible to produce high-quality papayas in our conditions, but the optimal harvest maturity stage in each season, along with the postharvest storage capacity, vary throughout the year, even with active climate control. Thus, 10% yellow skin color (S0) is the optimal for harvesting in autumn, mainly due to its better conservation capacity and firmness, having a high TSS content (>11 °Brix). In summer and spring, the lower TSS content discourages harvesting at S0, with S1 (30–40% yellow skin color) or S2 (50–60% yellow skin color) being the optimal stages for harvest to ensure the minimum TSS content required for papaya commercialization (10 °Brix). In any case, papayas produced under our conditions are generally well rated by consumers.