The use of protective covers, such as high tunnels, is recognized as an effective technology to reduce rain-induced fruit cracking in sweet cherries; however, there is a lack of information concerning the effects of this production system on the fruit’s mineral concentration, quality, and postharvest life. This study assesses the feasibility of using high tunnels on ‘Santina’ sweet cherries under the Mediterranean climate of the Central Valley of Chile to obtain earlier harvests of high-quality fruit with long storage life. The study included two plots: Plot 1 during the 2018/2019 growing season, and Plot 2 during the 2019/2020 growing season. High temperatures and relative humidity inside the high tunnels during bloom and fruit set decreased fruit yield, particularly in Plot 1. On average, trees inside the high tunnels were harvested 11 days earlier than those in the open. Fruit from covered trees were significantly larger (13%) and softer (10%) than those from the outside. Fruit quality characteristics, such as soluble solids concentration and titratable acidity, were not affected by high-tunnel-protected cultivation. Fruit from covered and uncovered trees maintained the firmness differences obtained at harvest between treatments, but showed similar postharvest quality after 45 days at 0 °C and a further 3 days at 20 °C on the other characteristics. The covered fruit had lower Ca concentrations (7.7 mg 100 g−1) and higher K:Ca, Mg:Ca, and N:Ca ratios. Significant relationships were found between Ca or K:Ca and fruit firmness at harvest. Lower Ca concentrations in the fruit may explain the lower firmness of fruit grown under plastic covers. There were no differences between covered and uncovered cherries in either cracking susceptibility or induced pitting. ‘Santina’ cherries were very sensitive to pitting damage, but this is not associated with the fruit’s Ca concentration. The results obtained show that high tunnels influenced fruit yield, development, and quality, and emphasize that the fruit’s Ca concentration under this growing condition plays a significant role in the firmness of ‘Santina’ sweet cherries.