How should we approach the authors of a paper and ask them for their data? Nowadays, many papers are based on data that can easily reach several Gigas. It is understandable that these data are not incorporated into the work, nor are hosted with the publisher. What is the more acceptable way of asking them to access this data? Call + email? Email to the publishers? Making a Freedom of information request to the university?
It's very likely the publisher doesn't have the data if it wasn't published as an online supplement to the article.  

You should start by contacting the authors (typically by email but perhaps by phone if email doesn't work) and asking for the data.  In my experience, many authors will happily give you access to the data, and if they are prepared to do this, a polite request will be the quickest and easiest way to get it.  Furthermore, it's most likely the case that you will need help from the authors in understanding details of the data.  If you've worked with them in a cooperative fashion, then there's a much better chance that you'll be able to get such questions answered.  

There are various reasons why the authors might not be able to share their data with you.  For example, in research projects involving human subjects research the IRB approval for the study might not allow for such a release.  Another common situation is that a company has sponsored the research and considers the data to be proprietary and the researchers are contractual obligations that do not allow them to release the data.