Monkeyman! / Buses, Bikes and Subways - S5-E1
Continuity mistake: Monkeyman!: When Sid tells the kids that Gerald should tell the legend of Monkeyman, Arnold is squinting. In the next shot, he's not. (00:01:41)

Monkeyman! / Buses, Bikes and Subways - S5-E1
Continuity mistake: Buses, Bikes, and Subways: Helga wears a watch on her right arm throughout the episode. However, when Helga approaches Sheena's uncle Earl and wakes him up, her watch is missing, then reappears when Earl's boat sinks. It disappears again when Helga comes up from the water, then reappears a moment later. (00:18:59)
Monkeyman! / Buses, Bikes and Subways - S5-E1
Continuity mistake: Mr. Simmons counts 30 students in the bus, but nowhere near that many got off (nor have nearly that many fourth graders ever been shown).
Helga's Masquerade / Mr. Green Runs - S5-E2
Continuity mistake: Helga's Masquerade: When Arnold compliments Lila's bows, she says, "In fact, I'm certain I wear them just, every old day." Ironically, her right bow disappears mid-shot, and it remains missing in the following shot. It comes back shortly after. (00:01:50)
Helga's Masquerade / Mr. Green Runs - S5-E2
Continuity mistake: Helga's Masquerade: When Helga and Lila are at Slausen's Ice Cream, after Helga asks Lila where she gets her material from, Lila's right arm is below the table, but a shot later, it's above the table, and a spoon appears in her hand. (00:03:18)
Helga's Masquerade / Mr. Green Runs - S5-E2
Continuity mistake: Helga's Masquerade: When Helga and Lila are at Slausen's Ice Cream, after Helga asks Lila how she can learn to act so nice to people, she grabs her milkshake with her left hand. A shot later, she is now holding it with her right hand, and a spoon appears in her left hand. (00:03:30)

Parents Day - S5-E6
Continuity mistake: During the flashback, when little Arnold says goodbye to his parents, in the next shot of Stella and Miles, the bag's strap is slung over his dad's left shoulder as a crossbody, but mid-shot, the strap switches to his right shoulder with the bag on the opposite side. (00:14:30)

Parents Day - S5-E6
Continuity mistake: In the scene where Arnold's parents leave for the last time, Grandma's hair is dark at first, then it becomes purplish-gray.

Parents Day - S5-E6
Continuity mistake: When Grandpa enters and exits Arnold's bedroom, the door opens the opposite way in various shots, with the doorknob on either the left or right. Additionally, the light switch beside the door appears and disappears, between shots.

Parents Day - S5-E6
Continuity mistake: In the tricycle section of the last event, the chinpiece of Helga's helmet keeps switching colors.

Helga's Locket / Sid and Germs - S5-E7
Continuity mistake: Helga's Locket: When Arnold is cutting a photo of Grandma in his room, there is nothing on his computer screen. Then, when Arnold starts to leave his room, a photo of Lila appears on the screen. (00:05:48)
Helga's Locket / Sid and Germs - S5-E7
Continuity mistake: Helga's Locket: When Arnold and Grandpa are in the basement trying to pry open the locket, the tools on the workbench constantly rearrange themselves in various shots. (00:10:06)

Helga's Locket / Sid and Germs - S5-E7
Continuity mistake: Helga's Locket: After Helga leaves the boarding house through the basement window at the end, when it shows a close-up shot of the locket in her hands, Helga's sleeves are way longer than they should be. (00:11:37)
Helga's Locket / Sid and Germs - S5-E7
Continuity mistake: Helga's Locket: When Abner takes the locket from Helga at the end, Helga chases after him. In the next shot, they are shown running past the basement window of the boarding house, but that's impossible because they started running away from the basement window in the previous shot. (00:11:42)

Helga's Locket / Sid and Germs - S5-E7
Continuity mistake: When Sid decides to be a slob again, he picks up his frog and you can see that he has gloves on. When the following shot shows a close-up of the frog in Sid's hand, the hand holding the frog does not have gloves on.

Summer Love (a.k.a. Beach Story) - S5-E9
Continuity mistake: When the seagulls start flying above Arnold and Summer, they are both holding hands. In the next shot, they're not. (00:11:26)
Summer Love (a.k.a. Beach Story) - S5-E9
Continuity mistake: When Helga sprinkles bread crumbs around Arnold and Summer, six seagulls come and fly above them. In the next shot, the shadows of the seagulls are visible, but there are only two seagull shadows. Then when both look up at them, there are only five seagulls flying above them instead of six. (00:11:26)

Summer Love (a.k.a. Beach Story) - S5-E9
Continuity mistake: When Arnold walks up to Helga after overhearing her, a surfboard suddenly appears behind Helga, which wasn't there before. A few shots later, it disappears. (00:11:42 - 00:12:07)
Summer Love (a.k.a. Beach Story) - S5-E9
Continuity mistake: When Arnold talks to Helga after overhearing her, he tells her that he heard everything she said. As Helga says, "You did?", the lifeguard stand that was behind her suddenly isn't there. Then when Arnold and Summer walk away from her, they are suddenly a lot further away from the lifeguard stand than they were before. (00:11:47 - 00:12:12)

Summer Love (a.k.a. Beach Story) - S5-E9
Continuity mistake: In the scene at the end where "Babewatch" is being filmed, when Arnold drowns and the lifeguards save him, he is wearing his plaid shorts. A shot later, he is wearing his blue swim shorts. (00:22:34)
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.