Factual error: The tanks, half-tracks and self-propelled guns are modern American vehicles painted in a a bright yellow color rather than a muted desert sand color. The only vehicles which actually did see action in North Africa in WW2 would have been the M-7 Priest Gun Motor Carriage and the M3 half-tracks being used by the Germans in "The Rat Patrol."
Tomorrow is Yesterday - S1-E20
Factual error: Towards the end of the show the Enterprise is leaving Earth orbit and heading towards the sun. We see the Earth diminish and the moon appear looking exactly as it does from Earth. From this angle we should be seeing the "dark side" of the moon, which looks completely different. (00:40:50)
The Clock King Gets Crowned (2) - S2-E12
Factual error: Trapped in the overturned hourglass, Batman and Robin lie flat and begin furiously digging in the sand, making the hourglass roll out the door. But to cause such rapid movement, they'd need to be applying their weight against one of the glass sides (like a hamster in a ball). Just digging in the sand wouldn't get them anywhere, at least not that quickly. (00:04:15)
The Jewels of Topango Affair - S1-E14
Factual error: Mark plays his electric guitar for the tribal chief, and the amplifier works just fine, despite the fact that it isn't plugged into any power source. (00:17:00)
The Legacy - S1-E15
Factual error: Reel-to-reel tape recorders were threaded left-to-right and pulled the tape in a L-R direction. The machine playing Briggs' assignment in the opening sequence, however, is running backwards. It also changes to a different reel and a different machine when he removes the finished tape from the wrong side of the recorder. (00:02:00)