(a) In those days… = その日…
(b) At that time… = その時…
Both (a) and (b) can be translated as その時代…, but I think その時代… is better used as a translation for "in those days…".
The example (b) can also be used for a very short period of time. For example, "at that time…" can be used for "on that morning..." or "on the way to school that day..." or "during class on Thursday..."