One key aspect of the idiomatic expression is that it is difficult or impossible to use productively. For example, take the 'normal' expression the hamster is approximately potato sized. From it you can produce many others: the hamster is approximately olive sized, the hamster is approximately truck sized and no rodent or humans would be perturbed (though some would probably require a dash). They all make sense and require no stretching of imagination to be understood.
But take I want the hamster to stay at arm's length from me. You can't say *I want the hamster to stay at leg's length from me or *I want the hamster to stay at Cadillac's length from me. OK, you would be understood, but the expression would sound peculiar, stilted, maybe even humorous.
That's why "at arm's length" is an idiomatic expression.
Idiomatic expressions are also usually connected with some specific locale, often very small. One of the ways to tell a foreigner is from his choice of idiomatic expressions; it the second language has been learned, as is usually the case, from a patchwork of sources and teachers, the idioms don't form a consistent whole.
expressions marked with a * are not linguistically correct. Thanks to liveforever for corrections.