What does 其 mean in this particular example?
不得其死:
不得其死,汉语成语,拼音是bù dé qí sǐ,意思是指人不得好死。也表示对恶人的诅咒。出自《论语·先进》
(from baike.baidu.com)
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不得其所 = doesn't get one's own (wished) place
不得其死 = doesn't get one's own (wished) death
It comes from 《论语·先进》, 若由也,不得其死然. That is 像仲由这样,恐怕会不得善终啊!得死 is 得以善终, thus 不得死 is the same as 不得好死, 其 is a modal particle for 恐怕.
In most cases it is similar to a literary version of 己. However in these type phrases I have alway taken it for the literary meaning of that/it(specific).
不得must not/unable to get 其that(specific) 死 death. I am not an expert on the evolution of chinese from clasiscal to present, so I can't shed any light onto how that(specifc) conveys the meaning of the one you want/desirable/good. The pattern shows in multiple sayings however.
Compare:
不得其死 unable to die that(good way/properly)
AKA can't die well. this is a way to curse someone
不得其法 unable to method/way that( good way/properly).
AKA can't use/learn a method well.
不得其門而入 unable to door enter that(good way/properly).
AKA can't find the proper aproach.
it is really hard to try to convey the character meanings in english when they are so different from what would be written in englsih or even regular chinese (my skills are not high enough!) I wrote a more natural english below to hopefully help make the pattern clear.