Below is some info I gleaned from another site. I cannot confirm the info below but I found it interesting if, in fact, is accurate.
There are a few ways to get a permit to own guns in Germany:
a) You can prove that you are in much higher danger than the rest of the society and need a gun for self-defense. IIRC, there are less than 20,000 carry permits in Germany, most of them probably in the hands of security services. Carry license = Waffenschein, "grey card"
b1) You want to become kind of an expert witness and need the guns to gain the necessary knowledge.
b2) You want to become a collector - difficult. You might get a permit for guns of value for cultural history. These permits cover a special "subject". I don't know much about this, but we have at least two collectors in our club, one collecting "handguns of the German police forces", the other collecting "German service rifles". AFAIK, you have to begin before WW1 and then collect your way until you reach contemporary guns. I doubt one would be granted a permit with the subject "handguns with polymer frames" or "semi-auto rifles in .223". Collector's license = Sammlerkarte, "red card"
c) You become a hunter. Now, hunting is a very serious matter in Germany!

Becoming a hunter in Germany in both time consuming and expensive. From what I hear from the hunters I know it takes about a year and costs a couple a thousand Euros.
d) You become a sports shooter. That's probably the easiest and most convenient way. You have to prove:
- A clean background, though I don't know how they do this in case of foreigners. But basically foreigners CAN get a license.
- Your practical and theoretical knowledge regarding guns and gun-laws. It used to be something like course taking two to four evenings with a multiple-choice test and a practical exam showing you can safely handle a gun, point out different calibers, etc.
- That you have been "regularly and successfully" practising for twelve months according to the rules of an approved shooting association.
An SOP could be:
- You find a club near you. There will be one, whereever you go, just have to look for them. There are more than 30 in greater Stuttgart e.g..
- Shoot as a guest. So you can see if the club fits your interests. Some clubs concentrate on different disciplines/types of guns/ranges.
- Become a member.
- Shoot about twice a month for a year.
- Apply for your license.
There are two licenses for sports shooters. The "sports-shooters license" ("yellow card") for multiple-shot muzzle loaders (mostly revolvers), non-semi-auto rifles, single-shot shotguns (including over-unders and SxS), single-shot pistols. The "green card" for any other gun: semi-auto pistols, revolvers, semi-auto rifles and shotguns, pump-action shotguns.
In both cases you have to prove your "need" for the guns. In case of the yellow card you only have to prove your need for the first entry in the card. In case of the green card you have to prove your need for every single gun.