"well it's the circumference divided by the diameter" is exactly what you say. You see there still are no "perfect measurements of circles" — how would you perfectly measure anything? In any case, what does it mean to know an irrational number "exactly"? You cannot know it to the last digit, because there are infinitely many digits. There are various formulas which describe pi exactly, which mathematicians usually take to mean that we know pi. These formulas can be used to calculate pi to as many digits as you want. The current record stands at 22,459,157,718,361 decimal places, but of course for practical purposes this is utterly useless. 39 digits are enough to calculate the circumference of the universe to within the width of a single atom.

Before Archimedes various approximations were used in place of pi, for example in Egypt (16/9)^2 was common (as used on the Rhind papyrus, a sort of ancient Egyptian mathematics textbook). Note that this is a fraction. The fact that irrational numbers (i.e. those that are not fractions) exist was a discovery of the Pythagorean school, and it was not known even in Archimedes time whether pi was irrational or a fraction. What he did was to calculate much better approximations, to see whether he could find out the exact value somehow. He could not, but he obtained approximations that were considerably better than anything done before, and what's more, he was the first one to give a general method of calculating such approximations. The ones before him must have been the result of measuring actual circles with string and guessing some fractional number close to the result.

So how did Archimedes go about doing this calculation? He started out with a regular hexagon inscribed in the circle. Each of its six sectors is an equilateral triangle whose side length is the circle radius. Thus the hexagon's perimeter is six times the circle radius.

Now cut the six circular arcs in their centers to form a 12-sided polygon. Then cut the 12 arcs again to get a 24 sided polygon. And again. And again.

In each step the polygon gets larger, and its perimeter gets larger, but it stays below that of the circle. This is because the polygons connect two neighboring vertices with a straight line, but the circle connects them with a curved line, which is longer — strictly speaking this fact needs to be proven, but Archimedes took it as obvious. Calculating the perimeter of such a polygon gives us a lower bound for the circumference.

Now do the same process with circumscribed polygons, i.e. polygons that lie outside the circle with their sides touching it. Their perimeters are larger than the circumference, which gives an upper bound.

So we get an interval of numbers inside which pi must lie. Using polygons with higher numbers of vertices (i.e. subdividing more times) makes this interval more narrow and thus your estimate of pi more exact. Archimedes subdivided the hexagon 4 times to get a 96-sided polygon, but it is clear that he could have done it more times if he had wanted to.

Now I said "we calculate the perimeter of the subdivided polygon", but how is that done? The first one (the inscribed hexagon) as we have stated has a cicumference of exactly 6·r (and the circumscribed one has 12·r divided by the square root of 3). The subdividing of each triangular sector into two new ones is a simple geometric operation using Pythagoras' theorem twice (which every Greek mathematician of the time knew). For the curious the formula is s'=sqrt(2r(r-sqrt(r^2 -(s/2)^2 ))), which gives the edge length s' of the subdivided polygon from the edge length s of the previous one.

Now these calculations contain lots of square roots. The way Greek mathematics worked they used approximations by fractions in all calculations (whereas we use a special type, decimal fractions). Archimedes would not say "the square root of 3 is about 1.7321", he said "the square root of 3 lies between 1351/780 and 265/153." Notice how again he did not give an approximation but an upper and lower bound. Giving suitably close bounds for the square roots involved he found that certainly 223/71 < pi < 22/7. There is a report by Heron of Alexandria claiming that Archimedes made even more elaborate calculations and found closer bounds than that, this was however not published or later became lost. The method would have been the same, at any rate.

Source: The Works of Archimedes, translated and edited by T.L.Heath (1897 Cambridge University Press)