"Without that" clause

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.

• 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 not in adverb clauses, at least not in Modern English.

• *When that I get back from vacation, we'll paint the garage.

But in Middle English it was OK.

• 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.

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.

At first I thought maybe this might be a Canadian regionalism, but apparently not. It's an archaism used by Cory Doctorow to convey exotic, other-wordly overtones. Here are instances from several centuries ago, back when it was current...

...ye shall never depart from this city without that I go with you. (note the "ye" - it really is old!)

...in 1463...no merchant...should sell wool...without that he accept "ready payment".

Today, the standard phrasing is "...unless I go with you". As regards the specifics of including the word "that" after "without", @John Lawler has provided the canonical answer.

The following article sheds a little more light on this question:

Recategorization of Prepositions as Complementizers: The Case of Temporal Prepositions in English

According to the author, without that is found "well into the 19th century (and dialectally into the 20th)". Certain Southeastern dialects of American English still permit constructions such as 'They never came to church without that they brought their bibles'.

As an aside it may be worth pointing out that without that, rather than the gerund, is the standard construction in languages such as German (ohne dass) and Italian (senza che).