Why specifically 3:38?
To me the question about size of the song depends a lot on the details. I find some small songs (Lorde's cover of Everybody wants to rule the world with 2:37 mins or The Beatles Words of Love (also a cover) with 2 mins) very enjoyable. Especially small instrumental songs, if they are very melodic, I think they can easily get away with being small (For the damaged coda by Blonde Redhead, 2:40 minutes or Bron-Yr-Aur by Led Zeppelin, 2 mins). However, me, for example, when I was starting to compose, I would often have a very nice intro or chorus, or chord progression of some kind, that sounded really nice, but that I wouldn't be able to expand. Then I would try to convince myself that it sounded more "artistic" being small and what not, but they ended up sounding better when I put the effort to expand them. I also experienced the opposite, wanting to have a longer songs just for the sake of it and just repeating parts and in the end settling with smaller versions.
You can always ask your friends and family, being careful to chose people who actually enjoy the style you are trying to use. I have found that the time of the song is something that people will almost invariably point out, whether because they think it is too short or too long, and then you can decide what do you prefer.
Edit: An example of a song people told us not to make bigger: