There "should be" a space between family name (姓{せい} / 苗字{みょうじ}) and the name (名前{なまえ}) actually.
This wiki page has no English version but just for your reference: 和字間隔{わじかんかく}
I have read this post (姓名{せいめい}の表記法{ひょうきほう}について) and found a nice answer.
In articles, writers don't use spaces in character / people names.
But in many (maybe all other) situations, we put that space.
I'm sorry I am not sure the exact reason or the story of that spaces, etc. But the advantages are pretty clear. It will be much easy to know which is family name and which is the name. As you know Kanji doesn't like English. And also, it is type of respect. We do specially to writing others' names.
But the sum up, we don't HAVE TO put the spaces.
Also, about spaces in Japanese writting. Young students sometimes actually put a space after Joshi (助詞{じょし}) (particle), to make the sentence more easy to read.