朽木 (xiǔmù) · rotten wood

The figurative meaning comes from Confucius in 《论语·公冶长》 (the Analects, Book 5): 朽木不可雕也，粪土之墙不可圬也 - 'rotten wood cannot be carved, a wall of dung cannot be plastered'. Confucius said this of his student Zai Yu, who slept during the day. Today 朽木 alone, and especially the chengyu 朽木不可雕, brands someone as unteachable.

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Literal use ('rotten wood') is plain everyday Chinese; figurative use is sharp criticism - saying it about a student or subordinate is harsher than English 'hopeless case' because it carries the Confucian weight.