Minimum $\chi^2$ fit of experimental data in absence of fluctuations of one variable I know how to make a minimum-chi square fit when I have two variables both affected by statistical error. But how do I perform the fit when one of them has no statistical error, just the uncertainty of the instrument (for example say I am using a ruler which carries a constant uncertainty of 1mm) and according to my model this uncertainty cannot be neglected?
I use Root by Cern to make fits. I assume it follows this procedure to minimize $\chi^2$ of the fit when both the $x$ and $y$ variables are affected by statistical error (picture taken from L. Lyons, Statistics for nuclear and particle physicists, page 139)

This hinges on the condition that both values of $x$ and $y$ are gaussian distributed, so that you can evaluate the $\chi^2$ value corresponding to a given choice of the fit parameters. But what happens when you have no info on the statistics of $x$ and you just know that each measurement is sistematically affected by some fixed uncertainty of the measuring instrument?
 A: 
how do I perform the fit when one of them has no statistical error, just the uncertainty of the instrument

The best resource for learning about how to handle analyzing and reporting uncertainty in science is the NIST technical note 1297 on uncertainty: https://www.nist.gov/pml/nist-technical-note-1297
In there they classify uncertainty in two broad category. One is uncertainty which is evaluated based on statistical methods and the other is uncertainty which is evaluated based on non-statistical methods. In your case, for the specific measurement of interest the uncertainty would be the second type.
The best place to obtain the non-statistical uncertainty is in the manufacturer's documentation for the measuring device. If there is no documentation available or if the documentation doesn't specifically list the uncertainty, then it might be reasonable to assume that the uncertainty is based on the smallest digit of a digital display or the smallest tick mark of a non-digital display. If so, then you would model a reading of 1.02 V as coming from a uniform distribution from 1.015 V to 1.025 V. The standard uncertainty would then be equal to the standard deviation of a uniform distribution with a width of 0.01 V.
When you have both types of uncertainty, statistical and non-statistical uncertainties, then the combined standard uncertainty is the square root of the sum of squares of the individual uncertainties. This is basically the propagation of errors formula.
A: This is not a trivial problem: indeed, $\chi^2$ goodness-of-fit tests assume Gaussian distribution of the residual errors, and would be affected by the rounding error (since it renders these distribution non-Gaussian). A brief google search for "Goodness of fit rounding" shows that there is no generally accepted procedure for dealing with such situations, e.g., see
About the effect of rounding on the properties of tests for testing statistical hypotheses
Sensitivity of goodness-of-fit statistics to rainfall data rounding off
Parametric approach
One could design a statistical model for the rounding error (e.g., the values are uniformly distributed between two scale marks and rounded with probability $p$ to upper value and with probability $q=1-p$ to the lower one), which would give the general distribution of residuals for which one can construct a statistical test (likely more complicated than the $\chi^2$, but doable with modern computers.)
Non-parametric approach
One could look for non-parametric tests for goodness-of-fit, that is the tests that do not assume a specific underlying distribution. I would not be surprised, if some people would still argue that the Gaussian assumption is not necessary for using the least squares test. E.g., see What are some of the most common misconceptions about linear regression?
Practical approach
Posting a question on Cross Validated (the Stack Exchange forum for statisticians) is likely to generate more qualified answers than in PSE.
