Of American-English origin, the phrase how long is a piece of string? is a response to a question that cannot be answered precisely, although a precise answer seems to be expected.
—Cf. also the phrases:
– time flies? you cannot: they go too fast
– silent like the ‘p’ in swimming
– why is a mouse when it spins? because the higher the fewer

These are the earliest occurrences of the phrase how long is a piece of string? that I have found, in chronological order:

1-: From The Lawrence Daily Journal (Lawrence, Kansas) of Tuesday 1^st September 1885:

Orange Growing in Florida.

Correspondence of the New York Sun.
“How long does it take an orange grove to come into bearing?” The question was asked by a northern man in an earnest, deliberate way, that was intended to evoke a candid reply from the orange grower to whom it was put.
“How long is a piece of string?” returned the orange grower.
If he had been disposed to attempt an answer he might have said truthfully that an oronge [sic] grove will “come into bearing” in from six months to 15 or 20 years from the time of starting it, and that whether the interval is half a year or a fifth of a century depends almost wholly upon the wish of the owner.

2-: From The Railway Age (Chicago: The Railway Age Publishing Company) of Friday 3^rd February 1888:

How much does it cost a mile to build a railway? This is an extremely indefinite question, and yet, in substance, it is repeatedly propounded as a matter of mere curiosity. How large is a piece of coal, and how long is a piece of string, would be fully as sensible questions as to ask the cost of a mile of railway where the topography of the line, character of work to be done, number and kind of bridges, weight of rail, number of ties per mile, and numerous other considerations are not specified.

3-: From the Linn County Clarion (Mound City, Kansas) of Friday 9^th March 1888:

A correspondent from Wall Street writes the Clarion to know “when men will cease lying.” Dear questioner, how long is a piece of string?

4-: From The Indianapolis Journal (Indianapolis, Indiana) of Friday 27^th November 1891—quoting The Mirror and Farmer (Manchester, New Hampshire):

How often do we hear the question asked: “How much does it cost to produce a pound of milk?” It is like asking: “How long is a piece of string?” As strings vary in length, so milk varies in cost.

5-: From Annual Reports of the Dairymen’s and Creameries’ Associations of the Province of Ontario 1892 (Toronto: Printed by Warwick & Sons for the Ontario Department of Agriculture, 1893):

A Member: How many tons do you get to the acre?
Mr. Gould: Well, how long is a piece of string? Your question is a perfectly fair one, but under no system of agriculture can a man be sure of raising so much per acre. If we get sixteen to eighteen tons of good ensilage corn per acre we are satisfied.

6-: From the column Varieties, published in the Columbus Evening Dispatch (Columbus, Ohio) of Friday 19^th July 1895:

New York is wrestling with the question, “What constitutes a meal?” It would be almost as wise to discuss, “How long is a piece of string?” Two unknown quantities figure in the problem—the amount of ready cash and the state of the person’s appetite.

Various jocular replies to how long is a piece of string? have been made up, in particular twice the length from the middle to the end and variants.

For example, the following is from the column The Other Fellow, by Ad Schuster, published in the Oakland Tribune (Oakland, California) of Thursday 21^st March 1935:

In the Farm Credit Administration applicants lined up for jobs and were given tests of a kind which some find so delightful.
“How long is a piece of string?” was one question. And another: “How far can a dog run in the woods?”
Now, the cute little trick is that you are supposed to write down “no answer,” and if you do anything else you miss.
But there was one fellow who replied, “a piece of string is twice as long as the distance from the center to either end”; and, “a dog can run only half way into the woods; after that he is running out.”
We don’t know whether this man got the job he was after or not, but we think he should have the one held by the expert who was asking the foolish questions.

The same joke occurs in Mutt and Jeff, a comic strip by Al Smith (Albert Schmidt – 1902-1986), published in the Fort Worth Star-Telegram (Fort Worth, Texas) of Wednesday 28^th April 1937:

Jeff Unbuttons a Couple of Wise Cracks

Think I’ll have some fun with the little dumbbell – I’ll ask him some trick questions that nobody can answer!
Jeff, you’re a pretty smart guy – How long is a piece of string?
How long is a piece of string?
A piece of string is twice the length of that part between the center and either end!

O.K. Answer this one – How far can a dog run into the woods?
A dog can run only half way into the woods, after that he is running out of the woods!
Any more?