Concept:The sentence uses the fixed correlative conjunction pair no sooner...than to connect two events that happen in quick succession.
Explanation:The sentence is: No sooner had we set off ___ it started to rain.
The expression no sooner is a negative adverbial that requires than as its correct partner.
In standard English grammar, no sooner is never followed by that, when, or as.
The structure no sooner...than shows that the second action occurred immediately after the first action.
So the meaning here is that the rain began immediately after the speaker and companions set off.
The inverted construction "had we set off" is also typical of this pattern.
Therefore, the only word that fits correctly is than.
Answer:C. than