Is there a way to determine the most similar variables from a number of different time-series?
Since it is a measurement problem, maybe what you want to look at is how similarly the instruments mismeasure the ground truth or actual phenomenon. That is, look at the correlation of the measurement error (defined as the measurement minus the ground truth) between pairs of instruments. 
If you create a matrix X which has 1 column for each instrument and 1 row for each set of measurement errors at a given time, then the correlation is just transpose(X) times X. If you don't have ground truth, maybe the mean of the measurements is a workable substitute. If you aren't measuring the instruments all at the same time, calculating the correlation between errors will be more involved.
Since this is mostly a discussion question, it's really more suitable for stats.stackexchange.com. Good luck and have fun, it's an interesting problem.
I have a time-series data frame with 27 different variables. In reality, they are all different instruments recording the same measurand every 5 minutes.
I want to know if there is a way of determining which (say top 5 most similar) of the variables are numerically most similar.
If I was to calculate this, I would calculate the average difference between the recorded values for every possible instrument pair for every 5 minutes and then find the 5 instrument pairs with the smallest average difference. Doing this manually would take a long time, as I would need to calculate approximately 27*27=729 pairs and then find the pairs with the smallest average difference.
Is there a better/simpler way of doing this?
I have looked into correlation, but this will not yield the result I want - this will show how correlated the variables are, not which ones are most similar numerically.
Hopefully, this makes sense.