Factual error: The professional hitmen murder their victim in the hospital with a silenced .38 revolver. You'd think the mob would know better. Silencers have absolutely no effect on revolvers. They only work on closed-chambered, automatic weapons.
Character mistake: Cameron's attempt to assassinate the governor fails because McGarrett has everyone wear bullet-proof vests. It's been established that Cameron is brilliant, a top-of-his-class ex-cop and a crack shot. And he's been sending death threats, so he knows they'll expect him. Yet he fires at the governor's back and is shocked that the man doesn't die. Anyone with his expertise would have anticipated the vests, and aimed for the man's head.
Audio problem: During McGarrett's second visit to question the professor, in the midst of their conversation, something off-camera makes a very loud thump. Professionals that they are, Buddy Ebsen and Jack Lord carry on with their lines as though nothing has happened. But this looks just a bit odd, because you wonder why their characters aren't curious about the noise.
Factual error: An injured McGarrett is trapped in an overturned car, which emergency workers flip over before cutting him free. No rescue crew would ever do this: such violent movement could kill the patient. It's crucial that crash victims be extricated without causing further trauma: McGarrett would be freed leaving the wreck exactly where it was - upside down.