"Dog Days" is set in the fantastical world of Flonyard, a realm where anthropomorphic animal-like beings live in peace but occasionally engage in non-lethal wars as a form of sport. These conflicts, known as "Wars," are overseen by the ruling princesses of the various nations and are designed to resolve disputes without causing harm to the participants. The story begins when Shinku Izumi, a middle school student from Earth, is summoned to Flonyard by Princess Millhiore of the Biscotti Republic. Millhiore seeks Shinku's help as a "Hero" to defend her nation against the invading forces of the Galette Lion Dominion, led by Prince Leonmichelle.

Shinku, initially bewildered by the situation, quickly adapts to his role as a Hero, gaining special abilities in Flonyard that allow him to fight effectively. He is joined by other Heroes summoned from Earth, including Nanami Takatsuki, a skilled swordswoman, and Rebecca Anderson, a sharpshooter. Together, they assist Biscotti in its battles, forming bonds with the inhabitants of Flonyard. The series emphasizes camaraderie, teamwork, and the lighthearted nature of the Wars, which are treated more as competitive events than actual combat.

As the story progresses, the focus shifts beyond the Wars to explore the relationships between the characters and the mysteries of Flonyard. Shinku develops a close friendship with Millhiore, while Nanami and Rebecca also form connections with key figures in the world. The narrative introduces additional challenges, such as the emergence of ancient threats and the discovery of new lands, which test the Heroes' resolve and deepen their ties to Flonyard. The series balances action with slice-of-life elements, showcasing the daily lives of the characters and their interactions in this vibrant, whimsical world.

The main cast includes Shinku Izumi, the earnest and determined protagonist; Princess Millhiore, the kind-hearted and resourceful ruler of Biscotti; Prince Leonmichelle, the charismatic and honorable leader of Galette; and Eclair Martinozzi, a loyal knight of Biscotti who serves as Shinku's mentor. Supporting characters, such as the mischievous courier Brioche d'Arquien and the stoic warrior Gaul, add depth to the ensemble. The series spans multiple seasons, with each installment expanding on the world-building and character development while maintaining its core themes of friendship and adventure.