I'm puzzled by the behaviour of distance words such as inch, foot and mile. Their use of plurals seems odd. For example, one would say "a three-inch nail" but it's "three inches long"; a twelve-foot wall is twelve feet high; a four mile run covers four miles. What is the rule here?

I'd venture to say this occurrence is not related to "distance." For example, we commmonly refer to basketball teams as five-man, though there are 5 men. A three-woman singing group has three women. In English, we have the ability to convert a plural-oriented adjective into a singular form. Wish I had a two-screen computer.

Hi Brandon,
Further examples are five-dollar bill and two-bedroom house. But your formulation of the RULE doesn't satify me. The rule should say something like "nouns preceded by numbers are put in singular if..." if what?
Best regards

Isn't it about the 'head' word? If run, wall, team, etc are treated as singular, then the adjectival forms for them - 4 mile, 12 foot, 5 woman, will simply agree with this singular.

While we may not see the particular 'logic' in this, it appears to be a consistent rule. Even when the head noun is pluralised, we seem to remember this rule - "There were 12 five-foot walls around the courtyard".

cheer

the sunshine warrior

The distance is a singular distance. Five foot, three inch, &c. The unit used to measure the distance is what is being pluralised. The nail has a single length, this length is not three units. The measurement of the nail is three inches. A five feet wall would be a wall of five levels each a foot in height. In the same way a five men side of would consist of a team of five groups of men of unknown size (but greater than one), a five man side consists of one side of five singular men. Completely logical and straight forward when you realise that a team of several people is one team and a length of several units of measurement is one length.

To add to Philip Davis' explanation, you can charecterize those hypenated words as resembling collections, in that they have a singular form. Instead, they are descriptive nouns or adjectivized nouns. (Am I coining new phrases here?)
So, in a collection of Nails, (computer geeks would call it a class) one has, among others, Three-inch nails and five-inch nails(sub classes). IF I look, I may find a three-inch nail (an instance of the subclass) measures just 2.8 inches long because of a manufacturing defect.

To use the other examples, a five man team may actually have 9 men including reserves, but only five of them will play in the three-legged race.

Am I killing it to death?

HSallim has expressed it well. The example given above of the 5 dollar note is another very good example. It is not five one dollar notes but one five dollar note, one of many notes of several values.