How are strings and cords sized?

It really depends of the type of string or cord, and even it's intended use. Where, when, and why a type of material originated can affect how it is traditionally "sized".

Thread/String - Size 30 is tiny (like heavy thread). With a special tool, you can wrap it 30 times around the tool to get an inch. Size 20 is twenty times, so it's thicker. Size 10 is string-ish. Size 5 is yarn-ish.

String/Wire - can sometime reflect how many wires of a certain thickness can be contained in a one inch diameter tube (think phone cable). Size 30 can fit more than size 20, so 30 is smaller than 20.

Cord/Yarn - WIth fractions of inches, that is the diameter of the cord. 1/8 is larger than 1/16. Ply is less important in size these days (since each ply can be of any thickness today). 4ply used to mean thicker than 2 ply, and plies were somewhat a standard thickness in spinning (but not in rope making). It is still important in strength (more ply, less breakage). Some cords and yarns are knit or braided ply, rather than twisted (and breaks even less often).

Yarn - can also be listed by weight (not how much it actually weighs, but terms like worsted weight which is average, bulky weight is large, baby weight is small, bedspread weight is even smaller). Weight derrives from inteded use.

The industry today is attempting to standardize the terminology and sizing because of this problem. If you live near a craft store, go take a look at the items to get an idea of what a size looks like.

How do I know this? I use size20 wire in my job (very much just a metal string). My wife uses size 10 crochet thread. My mother sewed, My grandmother knit. My father tied fishing flies. My daughter wants to do macrame braclets. My son likes boats. I've got way too many stringy things in my life :-)

This is a very complicated answer. It all depends on what it is made of. Cotton uses one system of measurement, and wool another. Various other materials may adopt one or the other.

Basically, the amount of yarn you can spin out of a pound of raw fibre was used as the standard. The skeins were of specific length. So a size 30 would mean 30 skeins of cotton of a certain thickness would be spun out of a pound of cotton. But then that is singles. All yarns are plied, so you need to look at the number of plys too.

Just learn what a size 30 feels like, and what a size 8 and so on. Then it will be easier to judge what you need. Lower numbers (2, 4, etc) mean a thicker yarn; higher numbers (30, 60, 80) mean really fine yarns.

Most of the yarns are measured according to width or thickness of the yarn. Some companies will use millimetres, and some will use inches, as you have done. There doesn't appear to be a standard way to measure these.