Anthocyanins are often associated with health benefits. They readily degrade during processing and storage but are also dependent on the matrix conditions. This study investigated how strawberry anthocyanins are affected by preservation technologies and a relatively protein-rich kale juice addition during storage. A strawberry–kale mix was compared to a strawberry–water mix (1:2 wt; pH 4), untreated, thermally, pulsed electric fields (PEF) and high-pressure processing (HPP) treated, and evaluated for anthocyanin stability and bioaccessibility during refrigerated storage. The degradation of strawberry anthocyanins during storage followed first-order kinetics and was dependent on the juice system, preservation technology and anthocyanin structure. Generally, the degradation rate was higher for the strawberry–kale mix compared to the strawberry–water mix. The untreated sample showed the highest degradation rate, followed by HPP, PEF and, then thermal. The relative anthocyanin bioaccessibility after gastric digestion was 10% higher for the thermally and PEF treated samples. Anthocyanin bioaccessibility after intestinal digestion was low due to instability at a neutral pH, especially for the strawberry–kale mix, and after thermal treatment. The storage period did not influence the relative bioaccessibility; yet, the absolute content of bioaccessible anthocyanins was decreased after storage. This research further presents that processing and formulation strongly affect the stability and bioaccessibility of anthocyanins during storage.