Are all numbers real numbers? If I go into the woods and pick up two sticks and measure the ratio of their lengths, it is conceivable that I could only get a rational number, namely if the universe was composed of tiny lego bricks. It's also conceivable that I could get any real number. My question is, can there mathematically exist a universe in which these ratios are not real numbers? How do we know that the real numbers are all the numbers, and that they dont have "gaps" like the rationals?

I want to know if what I (or most people)  intuitively think of as length of an idealized physical object can be a non-real number. Is it possible to have more then a continuum distinct ordered points on a line of length 1? Why do mathematicians mostly use only R for calculus etc, if a number doesnt have to be real?
By universe I just mean such a thing as Eucildean geometry, and by exist that it is consistent.

With regard to the OP's question Can there mathematically exist a universe in which these ratios are not real numbers? to provide a meaningful answer, the question needs to be reinterpreted first. Obviously the real numbers, being a mathematical model, do not coincide with anything in "the universe out there". The question is meaningful nonetheless when formulated as follows: What is the most appropriate number system to describe the universe if we want to understand it mathematically? 
Put this way, one could argue that the hyperreal number system is more appropriate for the task than the real number system, since it contains infinitesimals which are useful in any mathematical modeling of phenomena requiring the tools of the calculus, which certainly includes a large slice of mathematical physics. For a gentle introduction to the hyperreals see Keisler's freshman textbook Elementary Calculus.
