Could one measure a stick to an arbitrary precision by having its length estimated by enough people? I remember reading somewhere that the problem of exact time-keeping on ships could have been solved a lot earlier than it was if somebody would have had the idea of keeping time with a whole array of imprecise clocks - taking the average of clock-times would have given a precise time.
Using the same reasoning, could I measure the length of a stick to atomic precision by showing it to enough people and having them guess at its length?
 A: No, of course not. Yes, some people will overestimate and others will underestimate. Averaging would cancel out the bias to some extent, but there's no reason to expect it to cancel out the bias perfectly. We all have similar eyes and brains. We are all deceived by the same optical illusions, in the same way. We all have a shared cultural understanding of when you should round a number up versus down. Because of all these shared biases, the average of an infinite number of humans would be a number that is not exactly equal to the real length.

Ask an infinite number of humans which of the top two horizontal lines is longer. Well above 50% will say the second one is longer. (They're actually equal.) You can ask more and more people, but the survey results will not approach 50-50. A large sample size cannot mitigate a systematic bias.
A: No because none of them know the actual answer. The averaging process you describe only works if each estimate is of the exact answer plus noise.
Otherwise it is known as the "Emperor's nose" problem. Nobody can see the Chinese emperor's face so they ask a million peasants how long his nose is, they average the results, and since they have such a large 'N' the standard deviation is very low.
A: I believe you are thinking of the Central Limit Theorem.
The mean and variance of the averages of many measurements are better estimates of the precision of your measuring rule, but don't tell you anything about the accuracy of your measuring rule. Your measuring rule may be biased.
The Central Limit Theorem is a part of mathematics. IMO you should also consider your question from the practical implications too. For example, take a look at the methods we have used to keep the Standard Metre - historically it was an iron bar, then this was replaced by a platinum-iridium bar, and now I believe it is defined by the distance light travels, in a vacuum, in 1/299,792,458 seconds with time measured by a cesium-133 atomic clock. This method (using an interferometer and atomic clock) would be a very precise way to measure an arbitrary 'stick' (if you could a devise a way to apply it) but does not guarantee an absolute precision (zero variance) in repeated measurements of the stick. 
A: This will not work.
I'm going to use the standard error of the mean as the measure of the precision: $\mathrm{SEM} = \sigma_x / \sqrt{N}$. $N$ is the number of people you have make estimates of he length, and $\sigma_x$ is the standard deviation of the estimates that everyone makes of the length. The standard deviation of the sample is given by the square root of the differencee between teh mean of the squared measurements and the square of the mena of the measurements: $\sigma_x = \sqrt{\langle x^2\rangle - \langle x\rangle^2}$.
So let's plug some numbers in to get a ball-park guess of what you need $N$ to be. The largest atom is apparently cesium, with a diameter of $520\mathrm{pm} = 5.2\times 10^{-10} \mathrm{m}$, so let's use that as a target for the precision. If all of the estimates you get from people have a standard deviation of $1\mathrm{cm}$, then you need $5.2\times 10^{-10} = 1\times 10^{-2}/\sqrt{N} \Rightarrow N = (1\times 10^{-2}/ 5.2 \times 10^{-10})^2 \sim 4\times 10^{14}$. There are just over 7 billion people on Earth, or $7\times 10^9$, so you would need every single person on Earth to estimate the length of your stick about $5.7\times 10^4 = 57000$ times each.
Of course, that all holds if the standard deviation of all of those measurements is about a centimeter. I assume it will be much higher. But even if it's lower, you don't get much benefit: you have to reduce the standard deviation of all those measurements by a factor of 100 in order to lower $N$ by a factor of 10. In order to get $N$ down to a "reasonable" number like 1 billion, you need to use a proper measurement technique.
Also note that the $1/\sqrt{N}$ behavior only works if all the measurements have errors that are uncorrelated with each other. You will probably run into problems with that assumption within each individual's set of estimates. So you're relying on people eye-baling it, you're really limited to about 7 billion independent estimates.
A: This is highly unlikely. It comes down to bias and variance. Individual people of course will estimate with limited accuracy, whether just guessing, eyeballing, or using latest and greatest measuring technology. By itself that would not be a problem if people were unbiased estimators and their estimates were independent. All errors would then be variance errors and could be eliminated to any desired precision by using the estimates sufficiently large group of people. This is the main idea behind wisdom of crowds
The problem is that people almost certainly are biased. One example is biase towards easily representable numbers. If the true length was very close 0.5 cm, it is very likely that the estimates would converge towards 0.5 exactly in the limit of a large number of estimates. Using a measuring device will not help its precision is not sufficient for an exact measurement. Measuring devices also have this bias. If the true length is .501 then devices with 2 digits of both accuracy and precision will all measure .50. you will never get the 0.001 no matter how many measurements you make.
A: I guess this depends on what exactly we mean by "estimate". If estimate means making up some number that's one thing. But if it implies some kind of [visual] measurement that's another thing. It may be difficult to think about a human being producing any good "measurement" visually, so let's ask a question instead: Can an accurate measurement of an object be obtained by averaging results from many machine vision experiments (http://en.wikipedia.org/wiki/Machine_vision)? To answer this we'd need to dig into how the measurement is made, what limits the resolution etc. If the measurement apparatus is the same every time then the errors from individual measurements will probably be statistically correlated and averaging of the results would not help to improve the outcome. But possibly [with some thinking] one can organize the measurement process so that the errors are not correlated between the measurements. If that's the case then yes, averaging of the results will improve the accuracy with the number of samples N since the error of the average would go as $1/\sqrt{N}$.
A: The people making measurements don't know what the precise answer is so that they can make 'imprecise' measurements. I think you need to understand what 'error' exactly means. Read this explanation:
A measurement may be made of a quantity which has an accepted value which can be looked up in a handbook (e.g.. the density of brass). The difference between the measurement and the accepted value is not what is meant by error. Such accepted values are not "right" answers. They are just measurements made by other people which have errors associated with them as well.
Nor does error mean "blunder." Reading a scale backwards, misunderstanding what you are doing or elbowing your lab partner's measuring apparatus are blunders which can be caught and should simply be disregarded.
Obviously, it cannot be determined exactly how far off a measurement is; if this could be done, it would be possible to just give a more accurate, corrected value.
Error, then, has to do with uncertainty in measurements that nothing can be done about. If a measurement is repeated, the values obtained will differ and none of the results can be preferred over the others. Although it is not possible to do anything about such error, it can be characterized. For instance, the repeated measurements may cluster tightly together or they may spread widely. This pattern can be analyzed systematically.
The average of a number of measurements can give a more accurate answer only when those measurements are precise. Those measurements will have errors and the average will also have an error, but it'll obviously be lesser.
A: I believe the answer is no. 
Let's simplify the question a little by limiting the number of persons doing the measurements to 1. Of course if you show the same stick over and over again and if the person knows she is shown the same stick, she will be making only one measurement. This can be avoided by sampling different length sticks, but the problem will persist if the observer knows the number of the sticks and can easily identify each. So she can only start giving a distribution of measurements only if the sticks she is guessing the lengths become visually indistinguishable to her or below her measurement accuracy. In this case repeating the measurement would get rid of the statistical uncertainty but there will be a remaining systematic uncertainty which is due to finite resolution of the human eye, accuracy of her "mental ruler" of etc. Proper treatment of the final uncertainty requires studying how the systematic uncertainties are correlated and constructing the covariance matrix. For example for completely correlated errors -which will be the case for the same person doing the measurement, errors are added linearly rather than in quadrature. So averaging does not get rid of the correlated systematic errors.
I don't have time to formally show it now (or to think about it actually) but I believe repeating the measurement with many observers instead of one is very similar. I think the final averaged result will have a combined systematic uncertainty that does not go away.
A: You can get a more accurate measurement from multiple measurements.  But each measurement has room to damage the target, and you would not get enough measurements to get to atomic standards.  Most likely, you could get to micron measures over 1000000 people, assuming each measurement is to the nearest mill.  But as likely, most readers would report 1234.12 mm as 1234 mm, so you would loose out a good deal of error there.  
You would do better if every ruler is in different, but known units.
A: 
I remember reading somewhere that the problem of exact time-keeping on ships could have been solved a lot earlier than it was if somebody would have had the idea of keeping time with a whole array of imprecise clocks - taking the average of clock-times would have given a precise time.

Define precise . Precise for timekeeping in ships would be accuracy enough to know the position of the ship to within a kilometer? a hundred meters ? some number useful for navigation. The clocks would be set by the noon sun, to the accuracy available in the position of the sun. The clocks would have to be unbiased in their error, within errors as many slow and as many fast etc. So in the end  exact will be qualified by the statistical addition of errors, plus an estimate of the systematics of determining noon sun, and the systematics of the clocks. 
In physical set ups there will always be a +/- error and precise means precise enough for the query at hand.

Using the same reasoning, could I measure the length of a stick to atomic precision by showing it to enough people and having them guess at its length?

No. There will always be a systematic error entering any measurement which means that the statistical error can be made very small but the systematic will limit  the accuracy. Using people to estimate numbers increases the systematic errors because of the subjectivity of the definition of 1cm, or 1 inch (three barley corns) in their heads. That is why one uses instruments where at least the systematical errors can be estimated by measurements.
