Yes, there needs to be a minimum distance. However, it's not hard enforced like personal and village claims. If you try to build another realm too close or on top of another realm, you create a zone of conflict. I don't remember exact details, so I'll have to point you to patch notes or the wiki.
http://ringofbrodgar.com/wiki/Realmviewtopic.php?f=39&t=50591&hilit=realm%2A -- Note: there have been several changes to realms since introduction, but that should get you started (hint: forum search "realm" posts by jorb, Announcements only)
Oh, and should note that I'm sure the realm isn't that big. Nobodies is. There's not enough players in the game to build a realm that large and keep it up.
Opinions expressed in this statement are the authors alone and in no way reflect on the game development values of the actual developers.