It is a sentence, but it is not what traditional grammar calls a complete sentence. In that context, this is a sentence fragment. Instead of containing a complete independent clause, this sentence contains a phrase. In turn, that phrase happens to contain two complete subordinate clauses.
A quick web search easily supplies some of the context for this fragment:
Did she have a precursor? She did. . . Oh, when? About as many years before Lolita was born as my age was that summer.
From this, we can construct the formally complete sentence that this fragment represents:
[She had a precursor] about as many years before Lolita was born as my age was that summer.
The subject/verb pairing of the implied matrix clause is she/had. Even though this is the primary subject/verb pairing of the sentence, it simply doesn't appear anywhere in the sentence as it is written. That is more than enough justification to call this sentence a fragment.
The phrase itself is quite complicated. "Before Lolita was born" is a subordinate clause, with "Lolita" as its subject and "was born" as its predicate. "As my age was that summer" is another, with "my age" as its subject and "was [something] that summer" as its predicate. The correlative "as...as" structure lets the "[about that] many years" serve as the "[something]" which was his age at the time.
A structure beginning with "as many as" can be the subject of a clause:
As many as a dozen different examples leap to mind.
Here, "as many as a dozen different examples" is the complete subject of the complete predicate "leap to mind." One simple parsing is that "as many as a dozen" modifies "different examples", leaving examples/leap as the simple subject/verb pairing.
If we know both the speaker's age and Lolita's age, then we can solve the math problem. Let's assume that he's 36 and she's 12.
There is some number x that represents both how many more than 12 years ago Lolita's precursor existed and how old he was at that time.
His current age must be his age at that time plus that same number of years before Lolita's birth plus the 12 years of Lolita's life since.
36 = x + x + 12
36 = 2x + 12
24 = 2x
12 = x
When did Lolita have a precursor? If we've guessed the current ages correctly, she had a precursor about twelve years before this 12-year-old Lolita was born, when he himself was also 12 years old.