(All the condoms are faulty. Abed need to notice everybody)
Go, Abed, Go. before people sex one another!
The construction of that fragment is a little convoluted, so let's break it down:
The subject "you" is implied, as is normal with the imperative. I've indicated this with parentheses. The word before is a conjunction, and it introduces the clause people sex one another, which is in curly braces.
The clause people sex one another is a straightforward transitive clause using sex as a verb, which in this case means "to have sex with". The object of the verb is the reciprocal pronoun phrase one another, which indicates that the action is being performed mutually within a group of two or more people. (This is subtly different from the reflexive, which indicates that the subject acts on itself.)
I've never heard "sex" used as a verb before. Also, the sentence beginning with "before" is incomplete.
I would rephrase the entire post thusly:
(All the condoms are faulty. Abed needs to notify everybody.) Go, Abed, go -- before people have sex with one another!
-- before people start having sex!
Traditional grammar might analyse the clause as follows.
People = subject
One another = an elliptical parallel/parenthetical clause:
In this hypothetical elliptical clause, one would be subject and another object. This probably best reflects the origin of the phrase. JSBangs' very useful analysis, following modern linguistics, might be used by more traditional grammarians as well, because one another is a fixed, idiomatic phrase and the class of reflexive pronouns is well established as a suppletive class.