Green leafy vegetables are very challenging in terms of storage and preservation, while packaging in controlled conditions with the selection of appropriate polymer material is crucial for maintaining their nutritional value and quality. Various packaging materials have different gas and water vapor permeability as well as physicochemical properties that can create a specific environment inside the package, therefore affecting the chemical composition, sensory characteristics, and overall quality of packed leafy vegetables. Stinging nettle is an edible plant with a high antioxidant content, making it a valuable leafy vegetable. Therefore, this study aimed to evaluate the influence of four packaging materials (biaxially oriented polypropylene (BOPP), low-density polyethylene (LDPE), polyamide/polyethylene (PA/PE), and polylactic acid (PLA)) on the antioxidant content of packed fresh nettle leaves during 14-day storage. Ascorbic acid content was the highest after 6 days of storage, equally well preserved in all tested films, with an average of 86.74 mg/100 g fm (fresh mass). After 14 days of storage, the total phenolic content was best preserved when packed in LDPE. The content of caffeoylmalic and chlorogenic acids was the highest in LDPE after 6 days. In addition, leaves packed in LDPE after 6 days of storage had the highest content of all photosynthetic pigments. According to FRAP analysis, the antioxidant capacity was best maintained in LDPE (at the 14th day, the measured capacity was 43.61 µmol TE/g). This study shows that the type of packaging material (BOPP, LDPE, PA/PE, and PLA) and storage duration (6 and 14 days) have a great impact on the level of antioxidant compounds in the nettle leaves, where LDPE and BOPP can be highlighted as the most favorable for the preservation of total and individual phenolic compounds, photosynthetic pigments, and antioxidant capacity.