The riddle
What Has Three Feet but Cannot Walk?
Three feet that are a measure
Wikipedia defines the yard as an English unit of length equal to 3 feet, or 36 inches, and says that since 1959 it has been fixed by international agreement at exactly 0.9144 meter. A yardstick is a stick one yard long, so it holds three feet of length. The word foot here names a unit, and the usual meaning of walking feet is what the question lets the listener assume.
The same riddle in 1924 and 1863
In J. Gilchrist Lawson's 1924 book the question reads, "What is that which cannot run though it has three feet always?" and answers, "A yard." An anonymous book, Riddles and Jokes, printed in London in 1863, begins "Three feet I have, but ne'er attempt to go" and answers "A yard measure." Both old answers are the measure itself, so some tellers say a yard or a yard measure, and a yardstick is the same thing in stick form.
From foot to yard
The international foot is defined as exactly 0.3048 meter, which makes three of them exactly 0.9144 meter, the length of the yard. Twelve inches to the foot and three feet to the yard give the 36 inches mentioned above.
Where the word yard comes from
The measure takes its name from Old English gerd, used for branches, staves and measuring rods, according to Wikipedia. It is a separate word from the yard of a house, which is related to garden. The article also gives 1,760 yards to the mile, a figure that fits 5,280 feet divided by three.