"The subject of irrationals in general was for the Greeks a part of geometry rather than arithmetic, and necessarily so, because, for want of notation, an irrational of any sort could only be denoted by a straight line or a combination of lines. This is illustrated by Euclid's Book X on irrationals simple and compound; these are always irrational straight lines, and the whole treatment of the subject is geometrical. The first discovery of the existence of the irrational must, however, have been made as the result of arithmetical considerations or reasoning with numbers." -Sir Thomas L. Heath
I guess, what you are really asking when you say, "does the square root of two actually exist?", is can the ratio of two numbers equal the square root of two. The answer so far, appears to be no, hence the name, irrational (not a ratio).
