There are other CV contributors who have backgrounds in measurement who may be able to give you a better answer, but I can give you some initial thoughts. Later, I could edit or delete this if need be. The page you like to seems to me to be decent, but heuristic. The independence, or lack thereof, is a really big deal, and it simplifies our tack immensely that we can assume independence.
The second big issue to worry about is bias. Essentially, you have a probability distribution over potential measurements. What you want to know (be the case) is that the expected value of that distribution is the true value of that property of the object in question. For example, imagine that the ruler you use was mis-marked at the factory when it was manufactured such that all measurements are a little too small relative to the actual length. On top of that, there is still a small amount of error in the use of the ruler. As a result, the probability distribution of the resulting measurements would not be centered on the true value. Let us assume that there is no bias in your case, which again simplifies our task.
The last issue, I should think, is the shape of the probability distribution of the measurement errors. I suspect it is reasonable to assume that your measurement errors are Gaussian. That is, it is more likely that your errors are smaller, and less likely that they are larger; furthermore, that they are symmetrical about the true value. It isn't necessarily true that they are perfectly Gaussian, but I bet they are Gaussian enough for our purposes.
All of this is background, but it allows us to use well-understood theory to address your situation. Now, what happens if you have measurement errors that are independent, unbiased, and distributed as a Gaussian? It depends on what we're doing:
- Your estimated mean should be unbiased. That is, it is centered on the true value of the population mean, as your errors will tend to eventually cancel each other out.
- The standard error of your mean will be larger than it otherwise would be from sampling error alone. For a simple simulation, draw random samples from a population distribution, then perturb each one a little, and calculate the mean. Do this over and over, and examine the empirical distribution of the means; they will vary more widely than would occur from sampling error alone.
- If, instead of calculating a simple mean, you were to correlate your measurements with another variable (and assuming the population correlation $\rho\ne0$), then the resulting correlation would be attenuated. That is, it would be closer to 0 than the true value.