Final answer:
The United Nations was established in 1945, between the start of the Great Depression and World War II. It succeeded the League of Nations and aimed to be a more effective forum for international cooperation and peacekeeping.
Step-by-step explanation:
Based on the clues provided in the question, the organization that was founded before World War II and after the time of the Great Depression is the United Nations. This occurred in a year that was not a leap year (consisting of 366 days), and not in a year of the Olympic rings, which happen every 4 years. As the Great Depression began in 1929 and World War II started in 1939, the United Nations was established in-between in the year 1945. The United Nations was created as a successor to the League of Nations, which had failed to prevent the outbreak of WWII. With the atrocities of the war being a fresh reminder of the importance of international cooperation, the United Nations was supported by major world powers and designed to be stronger and more effective than its predecessor.