Learn Russian: Мы не пускаем посторонних после обеда. - We do not let unauthorized people in after lunch.

It is actually in the accusative plural because it is the direct object of пускаем (we let in). The word посторонний (outsider) declines grammatically like an adjective. Because it refers to living people (animate), its accusative plural form perfectly matches its genitive plural form.

Is the word посторонних a noun or an adjective?

Grammatically, it is an adjective, but it is used here as a noun (a substantivized adjective) to mean an outsider or an unauthorized person. You will often see it on Russian warning signs, like Посторонним вход воспрещен (No entry for unauthorized persons).

Why use the imperfective verb пускаем instead of the perfective пустим?

We use the imperfective пускаем because this is a general rule or habitual action: we generally do not let unauthorized people in after lunch. If we used the perfective пустим, it would refer to a specific, one-time future action (we will not let them in this one time).

Does после обеда literally mean after lunch?

It can literally mean after lunch, but после обеда is also the standard conversational way to express the timeframe in the afternoon. Notice that the preposition после (after) always requires the genitive case, which is why обед changes to обеда.