Dear Megri,
I guess that the import will depend a bit on the type of data base. You seen the MNIST does not come in a standard format. At the bottom of the page you mention they tell you how the format is and what you need to program to extract it.
The format for different data bases will be different. Sometimes you might want to read data from a website. You then can use import. See for example this article:
http://community.wolfram.com/groups/-/m/t/435736
Sometimes you might take data from an API see for example here:
http://community.wolfram.com/groups/-/m/t/344241
Some databases are easier to access than others. If you want to access data from the Office for National Statistics in the UK, it is actually quite easy. You head to the data section and download a file in a standard format, usually Excel, and after saving the you use a simple import. This command does this for some crime data:
data = Import["http://www.ons.gov.uk/ons/rel/crime-stats/crime-statistics/period-\ending-march-2014/rft-table-5.xls"];
You then will have to figure out what the numbers mean, i.e. read meta data or explanations within the dataset, and you are good to go.
Accessing different databases usually requires some form of manual work. I don't think that there is a general procedure available right now that does all of that automatically. In fact Conrad Wolfram has some nice ideas about how making data more accessible would transform society/democracy. One of the problems is that there is no common standard yet to make all these datasets immediately computable - so some work is required. There is a really interesting workshop coming up that show how to use the Wolfram Language to use external data. Also, from what I hear the new Data Science Platform might be really useful.
From personal experience I would also think that the SemanticImport function is really useful to work with external data. It worked very nicely for some projects.
I have used large data sets from NASA or gigabytes of data from police forces to analyse data. Once you have imported and cleaned the data all this is possible even for large data set.
Is there any specific data set you want to access?
Best wishes,
Marco