It's just archaic English grammar, like using thou or sayeth. Like English spelling, it's correct, for an earlier version of the language.
That is a complementizer that marks a Tensed Subordinate Clause in English. It used to be able to appear in any kind of tensed subordinate clause -- noun clauses, adjective clauses, or adverb clauses -- as long as they have a subject, and a verb in the present or past tense.
This is still true for noun clauses (object and subject complement clauses), and for adjective clauses (restrictive relative clauses and NP complements), but not for adverbial clauses like unless I go with you.
- Noun clause (object complement): She thinks (that) he is asleep.
- Noun clause (subject complement): That he sleeps soundly is fortunate.
- Adjective clause (NP complement): The rumor that he escaped is false.
- Adjective clause (relative clause): The man (that) you called yesterday can't come.
but
- Adverb clause: Whan that Aprille with his shoures soote, ...
(That is normally optional, though under certain conditions it is required. In the above, optional that is parenthesized. Probably it was optional for Chaucer, who was a poet and needed a handy store of optional syllables. Just like Cory Doctorow, at least in that respect.)
The that in Chaucer's Prologue is not a demonstrative that -- Which April? That April, or the other one? -- rather, it's filling the slot where one used to mark a tensed time adverb clause, right after the wh-word that indicates time.
And the same goes for unless, which introduces a counterfactual conditional clause, also adverbial. We just don't do that any more with adverbial clauses in Modern English, so doing it is an affectation. Which is pretty typical of Cory Doctorow.