Monkeyman! / Buses, Bikes and Subways - S5-E1
Revealing mistake: Monkeyman!: When Arnold tells the kids that Monkeyman saved him, as he says, "He went after the two guys who were trying to mug me," he moves his right hand, and it's layered over Helga's hair. (00:01:25)
Career Day / Hey Harold! - S3-E11
Other mistake: Hey Harold!: When Harold and Patty are dancing together, as Harold says, "They said I didn't apply myself," Patty blinks twice, and her right eyelid is not colored in. (00:17:00)
Career Day / Hey Harold! - S3-E11
Other mistake: Hey Harold!: When Harold's family is talking about Patty during dinner, as Harold's mother says, "We can show her those adorable baby pictures of you," her left eyelid is not colored in. (00:20:39)
Helga's Boyfriend / Crush on Teacher - S2-E14
Other mistake: Crush on Teacher: When the kids are playing baseball and the baseball hits Arnold, as he says, "I'm fine, Miss Felter," his eyelids are not colored in. (00:13:02)
Helga's Boyfriend / Crush on Teacher - S2-E14
Other mistake: Crush on Teacher: When Arnold and Gerald are walking to school and Arnold gets distracted by the birds and flowers, as he says, "Look at the flowers," his eyelids are not colored in. (00:14:37)
The Aptitude Test / Oskar Gets a Job - S3-E9
Other mistake: Oskar Gets a Job: In the beginning of the episode, Arnold and Gerald overhear Oskar arguing with a man in his room. In the close-up of Arnold and Gerald as Oskar says, "You're a big man, aren't you?", Arnold's right eyelid is not colored in. (00:12:17)
The Aptitude Test / Oskar Gets a Job - S3-E9
Other mistake: Oskar Gets a Job: On the second day that Oskar forgets to do his job, Arnold reminds him that he has newspapers to deliver. After Oskar says, "You won't believe what happened," Arnold blinks, and his right eyelid is not colored in. (00:18:56)
Revealing mistake: In Arnold's nightmare, when Bob points out that a stork is flying off their roof, his legs are incorrectly layered behind his cape. (00:12:50)
The High Life / Best Friends - S2-E10
Other mistake: Best Friends: When Arnold walks across the table during the fashion show, a small chunk of the spotlight vanishes mid-shot. (00:22:13)
The High Life / Best Friends - S2-E10
Continuity mistake: Best Friends: During the fashion show, in the close-up of Arnold showing off the antennas on his costume, the curtains that he came out of are missing. They are also missing in the shot after Arnold tells Gerald, "Just smile and move, okay?" (00:22:22 - 00:22:50)
Helga's Masquerade / Mr. Green Runs - S5-E2
Other mistake: Mr. Green Runs: When Green tells Arnold he can't be himself and needs to come up with some fancy words, Arnold is wearing a "Vote Green" button. Then, just as Green says, "I gotta get outta here," Arnold's button now reads "Voet Green" with Vote misspelled.
Runaway Float / Partners - S2-E20
Other mistake: Runaway Float: After Bob says, "Now this float has to scream 'beeper' at the top of its lungs!", as he walks away from Arnold and Helga, Joey has white skin, blonde hair, and an orange shirt. (00:06:31)
Arnold's Room / Helga vs. Big Patty - S3-E4
Other mistake: Helga vs. Big Patty: When Helga dreams about boxing Patty, as she steps back from her, she is standing in mid-air for about a second. (00:17:00)
Arnold & Lila / Grand Prix - S3-E13
Continuity mistake: Arnold and Lila: On the bus, when Arnold tells Lila that he doesn't like her like her, Lila is smiling at him. In the next shot, she is suddenly frowning. (00:07:01)
The Beeper Queen / Oskar Can't Read? - S4-E4
Continuity mistake: Oskar Can't Read?: When Oskar comes home at the end, Ernie and Mr. Hyunh are standing on Arnold's left. As they walk inside the house, they are on Arnold's right. (00:22:36)
Full Moon / Student Teacher - S4-E1
Revealing mistake: Student Teacher: When Olga says, "But, nothing, baby sister!" before giving Helga detention, Rhonda and Brainy disappear for one frame. (00:17:17)
Runaway Float / Partners - S2-E20
Continuity mistake: Runaway Float: When Arnold first sees the beeper float, he says, "How can they control that thing?", and Gerald is sitting on the curb to the right of Arnold. In the next shot, Gerald is suddenly behind him. Arnold is also a lot further away from the curb than he originally was. (00:08:16)
Love and Cheese / Weighing Harold - S4-E7
Continuity mistake: Love and Cheese: When Arnold and Lila's boat starts sinking, it starts to move backwards and hits a huge rock at the edge of the waterfall. When Lila almost falls and Arnold holds her up, the rock is gone. It reappears in the following shots. (00:10:07 - 00:10:40)
Love and Cheese / Weighing Harold - S4-E7
Other mistake: Love and Cheese: When Arnold accidentally kisses Gerald's hand on the bus, his left foot is missing. (00:02:00)
Arnold's Thanksgiving - S3-E14
Other mistake: When Helga tells Phoebe how much she hates Thanksgiving backstage during the play, the third time Phoebe says, "Helga," the background is of Helga's living room, not backstage. (00:03:38)
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.