We can use 'at' as a preposition to express a state or type of activity. To set off, 'at the run' or, alternatively, 'at a run', means to 'start off running'.
At preposition (CONDITION)
used to show a state, condition, or continuous activity:
a country at war
children at play
At (Cambridge Dictionary)
You have quoted a 2004 translation of Zola's 1885 French work. Here, for comparison, is an 1895 translation by Havelock Ellis, of the same section:
When their garments had been thrown over her shoulders he set out
running, supporting his burden with one hand, and carrying the two
lamps with the other.
We could go to the original French text, which (I am sure) some people may feel is off-topic, but if you, dear Nadirspam, are reading Zola in translation, you might be interested:
Quand on lui eut jeté sur les épaules leurs vêtements, il partit au
pas de course, soutenant d’une main son fardeau, portant les deux
lampes de l’autre.
Partir - to leave, set out, set off, etc
Au pas de course - at the double, very quickly, fast, etc.
If you know a little French, you may see why an English translator (2004) might see au and think 'at the', and select something handy ('at the run') but then again, as Ellis (who was Anglo-French) showed in 1895, one may be flexible.