CBOR uses variable-sized length prefixes. Strings zero to 23 bytes long require just one byte of overhead, after that it becomes two bytes for strings up to length 255, and 3 bytes of overhead for strings up to 65535. Above that, it requires 5 bytes of overhead, which is probably enough for strings up to at least a few hundred GB, though I didn’t test that far.
click to see how i empirically determined those numbers
$ python -c 'import cbor; overhead=0; print({ length:overhead for length in range(65537) if overhead < (overhead:=len(cbor.dumps("a"*length))-length) })'
{0: 1, 24: 2, 256: 3, 65536: 5}
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