For medical statistics take a look at the Medical subject guide page on health statistics.
These are free and paid databases, maintained by either commercial outfits or by international agencies like the UN, WTO, World Bank etc,
These are free datasets provided by governments
Sometimes the data you need is available for free on the country's statistical agency's website. Not sure where that is? Try the 2 lists below.
Sometimes a obscure statistic can be found in a diagram in a book or journal article. Do a search in Google Books or Google Scholar to check for this. Google Books might not give you access to the full book but limited preview mode might allow access to the diagram or graph. Journal articles often state where the data source is from, so this will give you a clue as well where to look for the statistics you need.