A shuttlecraft carrying a research team, commanded by Spock, is lost in an ion storm, rendering sensors useless. The shuttle is forced to crash-land on a hostile planet populated by large, violent creatures that would like nothing better than to kill Spock's team one by one. Meanwhile, the Enterprise conducts a desperate search for the team (sans sensors, it's a needle in a haystack), with time running out. Before long Kirk will be forced to abandon the search and proceed to a threatened colony in need of medical supplies.
So at last, here's a full-fledged character analysis of Mr. Spock. "The Galileo Seven" certainly isn't a standout science fiction outing, but so what? History has shown us that Trek's evolution was one that put emphasis on its dialog and characters rather than in revolutionary sci-fi premises. And this episode, the original Shuttle Crash outing (Voyager writers take note), is a perfect example of what makes Trek so enduring. The simplicity of having Spock and six other crew members stranded on a planet gives us plenty of time to study "Spock's first command." It's fulfilling to watch Spock engage in a logical approach to a survival situation—so logical and lacking in emotional intuition that the rest of his team nearly turns on him.
Of particular interest is the way his logic is so sensible if you think it through, yet it still doesn't work in practice. The most brilliant line: "Strange—step by step I've made the correct and logical decisions, and yet two men have died." Spock seems trapped in a paradox where succumbing to emotion may be the only solution—which it is, as evidenced by an act of desperation that he ultimately takes ... an act that itself could be rationalized as a logical one given the limited options. A most clever story.