What's the minimum number of datapoints in order to run a diff-in-diff?
Diff-in-Diff is a parametric model. A rule of thumb for parametric models is that you should have at least 25-30 observations (different authors might disagree but it is around 30) per independent regressor used in your model (see discussion in Verbeek A Guide to Modern Econometrics pp 36). However, note this is a rule of thumb and as Verbeek points out (my comments are in []):

Unfortunately, there is no unambiguous definition of what is ‘sufficiently large’ [talking about sample sizes to justify asymptotic properties]. In
simple circumstances a sample size of 30 may be sufficient, whereas in more complicated or extreme cases a sample of 1000 may still be insufficient for the asymptotic
approximation to be reasonably accurate.

This being said getting precise number for every individual case would be too difficult (it would involve Monte Carlo simulations) so often people  follow the rules of thumb unless they have suspicion something is awry.
Consequently, if you would just follow the rule of thumb in a diff-in-diff model with 5 independent regressors you should have at least 150 observations, in model with 7 independent regressors 210, in model with 10 independent regressors 300 and so on.