Audio problem: Right after Helga is standing behind the "ARNOLD" on the pavement, when she says "Hey Arnold", her lips move as if she's saying it slowly (this is fixed in the 4th season).
Continuity mistake: When the four boys form their line (just before the adults call out "Arnold"), watch Harold's tooth; it goes from the right side of his mouth (as you look at him) to his left side.
Other mistake: In the closed-captioning, when Arnold and Gerald "touch thumbs" (and you hear that "sproing" sound), the caption says "[glass shatters]"
Continuity mistake: The first time you see the line of girls, Rhonda's shoes are "low-cut", but just before Helga says "Move it, football head", the shoes suddenly extend much higher (like they normally do in the stories).
Continuity mistake: The pets OK sign on Arnold's house changes in different episodes.
Answer: The show is based in present day times. Plus most cars aren't old fashtioned, I recall seeing a limo and not to mention a cherry picker, etc.