Continuity mistake: Spock states that the Fesarius (Balok's ship) "must be a mile in diameter". Yet any one of the small spheres that make up the Fesarius dwarfs the Enterprise. If the Enterprise is about 300m long, the Fesarius would have to be around 6 km in diameter - considerably larger than a mile.
Continuity mistake: As Balok's 10 minutes are counting down, McCoy comes onto the bridge wearing a standard blue velour uniform. A few minutes later, when Spock says he can bring up Balok's image, McCoy is suddenly wearing his medical tunic with the different fabric and collar. Then when Bailey starts losing it, McCoy is back in the regular velour costume again.
Visible crew/equipment: After the Fesarius destroys the recorder marker, when Balok declares that Enterprise must also be destroyed, in the next wide shot two tape marks are visible on the floor, one in front of Kirk's feet and another near Bailey. (00:21:50)
Other mistake: When Kirk is in sickbay with McCoy he uses sickbay's desktop monitor to contact Spock on the bridge, then when Kirk is in his quarters he uses his own desktop monitor to contact Spock again. Both have the same paint dings and smudges around the console screens, revealing the same one was used during filming. (00:04:30 - 00:06:30)