Just before the quoted passage, the narrator says a whistling going "Don!" happens at the same time as he stumbles and ends up stooped. Then:
もちろん、いくら昔の話でも発車のベルは「ドン！」などとは鳴らないし、発車のベルが人を突き飛ばすこともない。
My attempted translation:
Of course, though this happened a long time ago, it couldn't be possible that the bell would go “Don!” and send someone flying.
Here I assume こともない means "it's not possible/the case that..." Is this correct?
「～こともない」 is made of 「～ことはない」(don't ever~~) + 「も」(either).
「～ことはない」 is the negation of 「～ことがある」(occasionally do~~).
Compare:
ベルが人を突き飛ばすことがある。-- The bell occasionally/sometimes pushes people away.
ベルが人を突き飛ばすことはない。-- The bell wouldn't ever push people away.
ベルが人を突き飛ばすこともない。-- The bell wouldn't ever push people away, either. / Nor would it ever push people away.
「ドン！」などとは鳴らないし、発車のベルが人を突き飛ばすこともない。
「～～ないし、～～もない」 means "neither ~~ nor ~~".
"The bell would never go 'Don!', nor would it ever push people away."
-「ドン！」
- 人を突き飛ばす "to shove people by the explosion of the round shot"
「ドン!」in the story of がばいばあちゃん, I think, is not a mere onomatopoeia of something, but the sound of a blank shot by a heavy gun or a cannon. In Meiji era they fired a blank shot to tell citizens the time of noon, which is written here. Sometimes ドン meant, I once heard, not only the sound of a cannon but a cannon itself in those day.
There is another ドン which is something to do with this ドン; that is used in 「位置{いち}に着{つ}いて、よーい（用意{ようい}）、ドン」 which means On your mark!, Ready! / Get set!, Go!. The last ドン is of course the sound of firing a starting pistol.
こともない means "it's not possible/the case that.
こともない is a kind of emphasis of "have/do not either."
It is emphasizing or pointing out that a bell would never go "don" or shove someone.
