According to Ed Robertson in 2018:

I had the basic structure of the song, the choruses, which is kind of this silly deconstruction of an argument between two people who actually really like each other. The kind of stubborn argument that drags out over a week. I had that, and I wanted to do some kind of rappy verses but they always sucked every time I tried to write them. It was finally Steve [Page, Barenaked Ladies co-vocalist] who said, “Why don’t you just freestyle it, like you do in every show? The freestyles you do off the top of your head are better than the stuff you’re trying to write.” I wasn’t very sophisticated at that time in terms of the whole home studio or recording equipment. So I set up a video camera and I freestyled like four verses and I edited those down into the two verses that became “One Week.” So our biggest single ever, our #1 single, was written in three-and-a-half minutes.

I thought it would be a bonus track or a b-side. It was one of the last songs I submitted to the record company and when Sue Drew, who was our A&R person at the time, said, “We wanna lead with ‘One Week,’” I actually thought she was joking. I thought she was making a dig at me, like this is the stupidest fucking song I ever heard. Which I would’ve agreed with. I labored over so many songs on that record, and tried to make them, you know, super deep and meaningful and soulful and tried to nail them emotionally, and then this totally ridiculous song that I improvised, that makes no sense at all, goes to #1.

According to Ed Robertson in 2018:

Absolutely not. It is a hodgepodge of pop culture references and inside jokes. I can tell you where every single line comes from and what it means, but they don’t relate to each other, and they don’t relate to the chorus.