I think originally the reason for these odd sizes is because bullets were ball shaped and were measured not by caliber but by the pound. Early muskets and flintlocks used spherical lead balls, which were measured by how many equal sized balls you could make with a pound of lead, much the same way buckshot or birdshot is measured today.

So, for example, a 2 oz ball (8 to the pound) is 8 gauge, or 0.835" in diameter, or a 1 oz ball (16 to the pound) is 16 gauge, or 0.662" diameter. When you get to smaller diameters, you are in the realm of buckshot or birdshot. For example, 0000 buckshot is 0.38", 0 buckshot is 0.32", #4 buckshot is 0.24", to name some common diameters for bullets used today.

Such spherical balls were used through the early centuries of firearms, in fact up to the early 1820's, when conical bullets were invented. But even much later, spherical bullets continued to be very common, such as in the widely used ball-and-cap pistols in the Civil War.

These early guns were not very precisely bored, so it can be deceptive looking at very precise numbers like 0.662" and thinking "wow, that must have fit a gun with a 0.662" bore". In fact, bore size only roughly approximated the bullet and accuracy was low at large distances.

After conical bullets were invented, there was no reason to stick to the old "how many balls per pound" method of sizing bullets. Some of the variation is for historical reasons. For example, there was originally the .38" bullet, that was designed for use in a black powder cartridge. Since black powder is weak, the manufacturers tried to squeeze in as much as they could, which they did by giving the bullet a very short "heel" or narrowed section that fit inside the brass. This left the bulk of the brass free to cram in lots of black powder. But later when smokeless powder was invented, this was no longer needed, so they seated the bullet more deeply in the brass - in fact, the whole bullet became the "heel". And it so happens that when you reduce a .38 bullet by the thickness of the brass, it becomes .357, which is a popular caliber today.

In the same way, a .44 magnum is really a .429" bullet that also reduced its size during the switch from black powder to smokeless powder. So it's not that someone decided "we're going to specify this bullet with 3 digits of precision because that makes for a great bullet", but rather, it was just a historical artifact of a change in process/materials.

Another thing to remember is that sometimes bullets were deliberately bigger or smaller than the designed bore for other reasons. For example, for a patched round ball the ball may actually be smaller than the bore, with the idea being that the patch makes up the difference. In other cases, the bullet may be slightly bigger than the bore, again with the idea that chambering it slices off a sliver of lead and ensures a tight fit. So you may have a bullet that's .002" bigger or smaller than the bore, and you wonder why it needs a 3rd digit of precision, did the manufacturers hate round numbers? But it has to do with the fact that the bore is already specified in round numbers, and now you are adjusting up or down from that for particular reasons.

I don't think there is any single answer for why such weird numbers for bullet calibers. It's basically a matter of long histories and the various changes that have happened during that time to manufacture, materials and technology, which have resulted in the actual calibers that are popular today.

The nominal dimension is pretty low on the list of priorities when designing a new round. Who cares if the bullet is 5.56mm vs 5.00? A smaller diameter bullet with the same proportions will have less mass, and therefore less stopping power. The reverse is also true.

The main calibers in WW2 for infantry rifles were mainly variations on .30 caliber. The US had .308, the Brits had .303, the Germans used 8mm, the USSR used 7.62x54R, the Swiss were prepared to use 7.5x55, etc. That is because .30 caliber had been in widespread use for decades and been proven for that application (main battle rifle).

As assault rifles became common after the war, countries moved to lower power rounds. This could be done by reducing the gunpowder (7.62x39) or reducing the bullet size and proportional charge(.223/5.56).

Once the application and general specs are decided, a large part of the decision is simply what is easily available. The .223 caliber bullet was in common use before the M16 was developed. The only development required was to choose a cartridge shape, powder charge, etc that worked best in the gun. The powder initially used in the M16's ammo famously caused some problems.

For highly specialized applications, lots of engineering and physics are involved. There is an amazing diversity of calibers, cartridges, and loads out there. Only a few are common household names because they get selected for high volume application, typically military.

EDIT:

Spend some time here and related lists to see how many different rounds share dimensions: http://en.wikipedia.org/wiki/7_mm_caliber

Details on the 5.56 round: http://en.wikipedia.org/wiki/5.56%C3%9745mm_NATO

Details on the 7.62x51 round: http://en.wikipedia.org/wiki/7.62%C3%9751mm_NATO

Not exactly the question you asked, but the response to How did some of the most popular sizes of rifle ammunition get chosen? might interest you!