Learn Italian: Nessuno entra nell'archivio senza l'archivista. - Nobody enters the archive without the archivist.

Because the negative word nessuno (nobody) comes before the verb (entra). In Italian, if a negative word like nessuno, niente, or mai is placed before the verb, you drop the non.

How would I say this if I put nessuno after the verb instead?

You would need to use a double negative by adding non before the verb. It would become: Non entra nessuno nell'archivio senza l'archivista. Both versions are perfectly natural and mean the exact same thing.

Why is it nell'archivio and not all'archivio?

The verb entrare (to enter) requires the preposition in (into/in), whereas verbs like andare often take a (to). Since we're talking about a specific archive that starts with a vowel, in combines with the definite article l' to make nell'.

Is archivista a masculine or feminine noun here?