Factual error: Don, who professes to be a Buddhist, chants in the jail cell and refers to Buddha as "god." Adherents to this religion do not worship the Buddha and never refer to him as god. Later on, Don chants to Krishna, a Hindu deity not even remotely connected with Buddhism. So, just what is Don - a Buddhist or a Krishna?
Continuity mistake: Danno, posing as a photographer on the beach, snaps the shutter at a bikini-clad girl and quips, "I only wish I had some film." Seconds later, he takes a picture of the suspects that's then used to identify them. The scene is continuous, and at no time did he ever load film into the empty camera, nor did he have another one.
Visible crew/equipment: 8-16 "Legacy of Terror": As an HPD cop escorts the old man out of McGarrett's office, he trips over a camera cable and nearly falls. When the actor recovers his footing and finishes his exit, the cast regulars simply pretend it never happened and go on with the scene.
Character mistake: After distracting the guards, Danno tosses a grappling hook onto a ledge and climbs the side of the building. He then goes about his business, carelessly leaving the hook and rope dangling from the ledge for all to see. For a seasoned 5-0 team member, that's pretty sloppy covert operations.
Plot hole: McGarrett is hunting for heroin dealers who've laced the drug with arsenic. Yet when he and Chin Ho arrest Harry Parch and find his stash, McGarrett performs the TV cop cliché of sticking his finger into the white powder and tasting it. Even that small a dose of arsenic could be lethal, and seasoned cop that he is, McGarrett would know that.