A fat notion includes properties. To say "x exists" is to say that, eg., it has a spatial and temporal location. This is often just baggage brought in from common ways of speaking and thinking.
It is more helpful to reduce "exists" down to its barest sense and discard any properties that often come along with it.
In this "thin" sense, exists only means that you can -- roughly -- successfully talk about it.
"Santa" exists, only that the object "Santa" does not possess the properties of being a man. He is a fictional object.
So "Santa, the man who visits you on christmas," does not exist. I cannot successfully talk about it (ie., i cannot refer to that man, I cannot point him out, there is no way of "locating" him).
In this sense I cannot see how one could object to the existence of the square root of 2. It can be successfully pointed out in the same way any number can be (esp. see the article). Only, say, that it failed to possess some properties of interest.
I guess the hypothetical objector in this article wants numbers to possess the property of "being easily exemplified by stones" (or whatever), in which case sqrt(2) doesn't have that property.
Why it should be that "2" exists because it can be exemplified by fingers, and sqrt(2) doesnt because it cant be -- is a bit strange.
But history is full of people wishing to arbitrarily define a set of properties that count as existence.