Without legs, or (informal British) extremely drunk and unable to walk properly.
Simple combination of 'leg' (from Old Norse 'leggr') plus the suffix '-less' (from Old English 'leas,' meaning 'without'). The slang meaning for drunk emerged in British English, possibly because drunk people stagger as if without stable leg control.
The '-less' suffix is one of English's most productive affixes—you can attach it to almost any noun to mean 'without that thing,' which means English speakers invented new words constantly ('mapless,' 'carless,' 'smartphone-less') long before modern technology actually existed.
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