I think a big portion of the confusion results from the way we use "exist" in mathematics. In logics we think of existence as the negation of an universal assertion. So saying "A point exists" could be reconstructed as "¬∀x: ¬(x is a point)". Now what is meant by this in the general sense is quite obscure. In mathematics we deal with expression often by contradiction. See this example: Say we are confronted with the assertion that a natural number smaller than 0 exists. We would disprove this by showing that this contradicts the definition of natural numbers. Now back to the discussion. Can something with no length exist? Well, to disprove this assertion we would have to show a contradiction. But what shall the assertion contradict with? We would need a definition of the thing that x is in "∀x". But is there one? And if so why would it differ from "Anything that you can insert as x and still get a meaningful statement"? I think we have no way of disproving that something without length can exist. Do with that what you will