When Does the Animated TV Show "Spider-Man: The New Animated Series" (2003) Take Place?

"Spider-Man" (retroactively released as "Spider-Man: The New Animated Series") is a superhero animated television show showran by Audu Paden [adapting "The Amazing Spider-Man" #6 (1963) by Stan Lee and "The Amazing Spider-Man" #9 (1963) by Stan Lee as well as "The Amazing Spider-Man" #265 (1985) by Tom DeFalco] and released July 11th, 2003. 

It takes place on an alternate timeline of the fictional Marvel Cinematic Universe in the year: 

2003 AD 

We know this because, in episode 8 "The Party", a character dies and their tombstone reads: "1982-2003 BELOVED SON". Then, in episode 12 "Mind Games, Part One" (the first half of the final story), a different character dies and their tombstone reads: "1984–2003". 

Though the earlier stories aren't as clear about the calendar year, in episode 1 "Heroes and Villains" Peter Parker/Spider-Man refers to himself as "a college freshman" and he and Mary Jane Watson are shown attending Professor Williams' chemistry class. They appear to be still taking the same class in episode 8 "The Party" and she (at least) is still going to this class and studying for its midterm in episode 11 "When Sparks Fly". So (unless Mary Jane failed Chemistry and is re-taking the class) the entire single season spans less than one college semester's worth of time, mostly likely the first semester of their freshman year, which is about 4 months long– between September of 2003 AD and December of 2003 AD. 

Although it was first developed as an adaptation of "Ultimate Spider-Man" comic-books, this show was intended to be an extension of the universe established in the film “Spider-Man” (2002). The later Spider-Man films entirely contradicted elements of this show, forcing the material into the status of a new alternate universe.