I am going to asume that by "barbarian" you mean non roman Europe roughly during the time of the Roman Empire. Barbarian is a very unprecise and inpractical term here though. Furthermore, many of these so called barbarian societies were very much settled and at some places people even lived in large villages/towns or small cities.

I study archaeology in Sweden and can tell a bit about how Iron was procured in Scandinavia during the Iron Age and a bit about the prehistory of mining in the region.

Regarding the Iron: I was actually present at an axcavation of Iron Age settlements (ca 200-700 AD for these specific sites) at a town called Vårgårda where we found so called Bog Iron (swedish: myrmalm). Bog Iron has also been found at earlier excavations in the vicinity. This Bog Iron consists of rusty lumps found on the bottom of lakes, swamps and bogs. You could "fish" for this using some sort of long hafted scooping devices, or depending on depth, pick it up with your hands. I know this is/was quite common in Sweden but I do not know the geological process behind the Bog Iron ending up in the lakes and bogs. But in this case you actually don't necessarily need a mine to get access to iron.

Regarding mines: mining in Scandinavia, south scandinavia to be precise, dates back to the Neolithic (4000-1700 BC). These early mines were for the minig of flint stone. Some of them were simply large pits, and others actual small shafts through the bedrock, reaching a depth of about 5-6 meters. In Scania, southernmost Sweden, large areas of these kinds of flint mines have been found, at Södra Sallerup for example.

I am sure others can provide in depth answers regarding iron mining in other parts of non roman Europe but I hope these where som helpful examples. A great book for reading more about scandinavian prehistory is "Ancient Scandinavia" by T. Douglas Price.