Question: I am doing a study in which I am measuring a set of lengths using two different methods. I want to test how close the measurements given by the two methods are for corresponding lengths. Is there a way to do this? I know that a matched pairs t-test can give a p-value for the means being different, but how can I test how close the differences are to 0? 

In other words, since the two methods are measuring the same lengths, they should most ideally give the same measurement for any given length. I want to test how close this is to being true and quantify the amount of error that is present. So say I plotted the lengths given by one method against those given by the other. If they did give the same measurement for every length, it should form the line $Y=X$. Basically, I want to test the hypothesis that the data does form the line $Y=X$, calculate the mean error (maybe the average distance from $Y=X$), and perhaps produce a p-value for it.

Answer: 
I think you are mixing different tools and different goals. I summarize several techniques available for what I understand you want to do:

* t-test for paired samples tests the hypothesis that both mesures have the same mean and gives a p-value about it.
* Confidence intervals estimate an interval where the true difference in means is expected to be with a given probability (usually 95%). That seems what comes most close to "test how close the measurements given by the two methods are for corresponding lengths". Most statistical packages produce confidence intervals when doing a t-test.
* If you are interested in measuring the errors between the two measures (an not just the difference of means), standard deviation of differences or [coeficient of variation](https://en.wikipedia.org/wiki/Coefficient_of_variation) might be useful. Studying the distribution of errors and checking for normality can give you interesting information, too.
* How are the two measures related, or how are they related to error might be interesting too. You have plotted one measure against the other, but next steps could be plotting error against one measure (specially if errors are much smaller than measures, as usually happens) and performing a regression analysis.

Probably my answer is more a menu of things you can answer now than a real answer to your needs, but if you use it to make your (next) question more specific I hope we will be able to help you more.
I believe the simplest way to answer your question would be to make a scatter plot with the following bivariate data for each measurement method you wish to test: x=actual length, y=measured length by (insert method here). If your method was perfect, there would indeed be a y=x least squares line with a sample correlation of r=1. If you want to test whether or not one method is significantly better than another, you could test the significance of differences between the two competing methods via their scatter plot correlation coefficients using Fisher's r-to-z transformation. You can find the probability that the differences in measurement methods would occur randomly using the formula in this article: <http://www.statisticssolutions.com/comparing-correlation-coefficients/>.