There isn't . . . . . . . . airport near where I live. . . . . . . . . nearest airport is 70 miles away.

We use indefinite articles before singular nouns that are unspecific. Here, 'airport' in the first sentence is unspecific and begins with a vowel sound. Hence, it will take 'an' before it. We use the definite article 'the' before nouns that are specific. In the second sentence, we are referring to the specific airport that is nearest to the speaker's home. Hence, it will take 'the' before it.