It's just archaic English grammar, like using thou or sayest.
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.
(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.
A more modern (though regional) example is provided by Andy Griffith's comedy monologue, What It Was, Was Football, delivered in the comedian's native North Carolina dialect, which includes a number of archaizing features, including using that in tensed adverbial clauses.
About 12 seconds into the recording in the link above, Griffith says
- Different ones of us thought that we ought to get us a mouthf'l to eat
before that we set up the tent.
Since unless also introduces tensed adverbial clauses (a counterfactual conditional clause in this case), the rule would apply to it, too. In this dialect. But outside Modern English dialectal speech, it's pretty rare, so doing it in print is an affectation. Which is pretty typical of Cory Doctorow.