Why is it that, in ancient Greece, mathematicians tried to solve geometrical problems using compass and straightedge only and, apparently, only if that failed, they tried to use other tools? Note that this question is not about what they were (and weren't) able to achive using compass and straightedge alone, but about why that was their first choice.
The real reason is probably that straightedge and especially compass are the simplest, most primitive instruments, and also easy to make. At the same time they are quite accurate. (Straightedge is not so easy to make and is not so accurate, but certainly it stands next to the compass on these criteria). A modern high school student tends to think that to double the cube it is easier to measure the side, then to calculate the cubic root approximately. But this is anachronism. Deprive the student of all modern tools, and prohibit to use the decimal system. How will s/he do the job?
If you think of the tool which is needed to double the cube (a parabola, invented by Menaechmus, exactly for this purpose), it is much more complicated than a straightedge and a compass.
This real reason was disguised: these instruments were probably considered "sacred" at some time. There is a popular theory that in general, geometry has ritualistic origin:
For example, the problem of doubling the cube clearly has some ritualistic origin: the original question was about doubling an Apollo altar. And it is natural to expect that only some sacred tools were permitted.
At a later time the sacred principle became a philosophical principle: the straight line and circle are the only "perfect" figures because "they look the same at all of their parts." (In modern language: isometry group acts transitively on them.) This was explicitly proclaimed by Plato and Aristotle. In particular, Plato declared that the main goal of astronomy is to reduce all celestial motions to uniform circular motions. (Why circular? Why not on ellipses/parabolas?) This philosophy actually slowed down the development of astronomy.
Remark: the ancient Greeks were great theorists in mathematics but not so good in practical applications. They probably thought a straightedge was a great instrument without trying it in practice for precise constructions. That a straightedge is not really useful when high accuracy is required was only discovered by the 18th century by precision instrument makers. If the Greeks had discovered this, they would have limited their geometric constructions to "compass only." In fact, one can do with just a compass everything which one can do with straightedge + compass. That the straightedge is really redundant was proved in 19th century. See Division of the circle and compass constructions for a discussion of this.
The Greek appear to have been deeply concerned with the problem of change-and- permanance, what we call today 'invariants'. The (straight)line and the circle are the two obvious cases when an object moves and stays at the same place. Actually mechanical physics relies on the two when it admits as basic translations and rotations. Greek geometry is just thinly disguised mechanics and it has already caught this fundamental idea. Every form can be reduced to its permanent constituents which are just straight lines and circles.
Actually the Mohr–Mascheroni theorem showed that straightedge and compass constructions are possible with just a compass.
The Greek historian Herodotus writes of the origins of geometry in Egypt:
They say that that king [Sesostri, ca. 2000 a. C.] distributed the land among all of the Egyptians, each one having an equal lot in the a square shape, and from these subdivisions obtained tribute, having imposed an annual payment. If the river bore away a part, the owner announced the loss, and officials were sent to observe the extent to which the plot had been diminished for the purpose of adjusting the tribute payment. It is my feeling that this indicates the invention of geometry here, prior to its passing to Greece.
Geometry - literally earth measurement - was the responsibility of the "arpedonapti", those who knot ropes.
It is by tightening ropes that they drew the two simplest and most important lines in geometry: the straight line and the circle. The first, simply tightening a rope between two points, a kind of operation which image is still present in the expressions "to draw a line", "to draw a perpendicular"; the second, making one of the two points turn around the other which is held fixed.
So the suggestion is that the "earth measuring arpedonapti" of ancient Egypt used only lines and circles and this was the art inherited by the Greeks. (I'm not entirely convinced.)
Euclid makes no reference to the edge and compass in his axioms.
Source : The Garden of Archimedes.
The basic reason is that the straightline and circle are the most basic of shapes. Aristotle theorises about them in his Physics, for example.
Its still the case today. Except of course the idea has ramified into many new and different ways. For example, when we think topologically, the straight-line is the standard example of a non-compact space and the circle, the standard example of a compact space.