An illness so severe that one cannot be cured
Refers to situations or people that have worsened beyond the point of repair or helping (figurative or literal)
Negative
汤姆很擅长理科，而他的兄弟绝对是不可救药 | Tāngmǔ hěn shàncháng lǐkē, ér tā de xiōngdì juéduì shì bùkějiùyào
Tom is very good at science, but his brother is absolutely hopeless
[Source]
他虽然病得很重，但并不是不可救药 | Tā suīrán bìng dé hěn zhòng, dàn bìng bùshì bùkějiùyào
His illness is very serious, but he can still be cured
[Source]
None
无可救药 (wú kě jiù yào)
None
《诗经》(Shī Jīng), or "The Book of Songs." This Spring and Autumn Period (770 - 476 BC) collection of poems is considered one of the five Confucian classics
周厉王 (Zhōu Lì Wáng) was one of the kings of the Zhōu Dynasty (1046 - 256 BC). He was an exceptionally cruel ruler, and many other members of his court were hugely corrupt, and so the people of the land suffered greatly.
However, not all of the court officials were corrupt. 凡伯 (Fán Bó) pleaded with the king to take his responsibility as a ruler more seriously, but the other court officials mocked him repeatedly, and he was ignored. This devastated 凡伯 so much that he sat down and wrote a poem criticizing the court, which included the phrase 不可救药.
Later, the people were unable to continue withstanding the government abuse, and they rose up in such a fierce revolt that the government collapsed.