Although seemingly set in Japan, the inhabitants of the towns around Akademi High School, and even in Tokyo, all speak in English, suggesting that the schools attended by the main characters might be international schools for large expatriate communities in Japan. The food products in the local store seem to originate in Japan, but the first episode of the show depicts Sam spending one dollar bills issued by "The Japanese Government" (although these appear to have a value of 10 Yen each).
In episode 77, some of the characters discuss "World War 2.5", in which neo-Nazis fought, suggesting that the series is set in an alternate timeline to the one the viewers live in. As Rowan recounts carrying Silly as a baby during this war, it presumably occurred around the turn of the millennium, possibly as part of or in place of the Yugoslav Wars that happened in our timeline. As another clue, when Yuki chases after Sam with her knife after prom, Rowan comments "ITS LIKE 99 ALL OVER AGAIN", suggesting he may have been involved in the Kosovo War. (Possibly relatedly, Teacher VonHeimer mentions fighting "in a war to kill your kind" referring to the fish people, suggesting that the existence of fish people in that universe may have had geopolitical ramifications).
The first episode also depicts a calendar on the wall of the convenience store, showing that it is January, and the month/year begins on a Tuesday. (Monday the 14th is colored red to indicate Coming of Age day [seijin no hi], a national holiday in Japan). Some years which start on a Tuesday are: 1991, 2002, 2013, 2019, and 2030. In Japan, the school year begins in April, and the final term starts in January, usually in the second week of the year (with the first being part of the New Year holiday). One interpretation, therefore, is that the series starts on January 7th, 2013. However, later in the episode, Sam and Taurtis talk about people receiving homework "over the summer" and attending "summer school", which suggests that the calendar may be out of date.
In the second season, which follows directly on from the first, calendars can be seen which give the current date as approximately "That One Time 2027". This can be seen for example in the office of Mr Nautilus in episode s2e11. The only month in 2027 starting on a Saturday (as on the calendar) is the month of May.