As far as I know, you can't own a sword in Japan without a license. Those licenses are only granted for historically or culturally significant swords. This would naturally make a katana the easiest one since they are culturally significant in Japan. However, it's still a hassle many wouldn't bother with.
Years ago a friend of mine had some visitors from Canada, one of whom was a Japanese person who moved to Canada for work. I got an idea and took my katana to the friend's place, knowing all those draconian Japanese laws. The Japanese woman was quite delighted, having photos of herself taken, with the sword in her hands. Apparently she had never had the chance in Japan to touch one. Now, they do sell souvenir katana in Japan, but those are things that only look convincing from 10 or 15 meters away.
You can get a license to have a katana, but you must register with the police and have a valid reason (such as being a student at a dojo or a sensei). The police will contact said dojo to confirm your story, and they will take measurements of the sword and scabbard; any changes require updating the details on the license. And as far as I remember, you still have to go to the police station yourself in order to renew your license.
EDIT:
I dug a little deeper and got confirmation, because my memory was spotty.
Katanas are actually registered by the Board of Education, and the registration card has to stay with the sword, and is what contains all of the details of the katana, including its dimensions.
Going to the police is for a
keitai kyokashou (carry permit), which is the permit to actually move around in public with a sword, however this is not that heavily enforced as long as the police can verify why you need to be out and about with a katana (like I said earlier about being a student or sensei at a dojo). But getting it helps make things smoother with the police.