If you want to do your own thing, similar to this site, there is nothing to prevent you from doing that.
Plagiarism is related to copying someone's written words and calling them you're own (like a book or article). I think you are thinking of copyright laws.
Here's what the copyright office has to say about online work:
Copyright protects original authorship fixed in tangible form. 17
U.S.C. sec. 102(a). For works
transmitted online, the copyrightable
authorship may consist of text,
artwork, music, audiovisual material
(including any sounds), sound
recordings, etc. Copyright does NOT
protect ideas, procedures, systems, or
methods of operation. 17 U.S.C. sec.
Basically, as long as you don't steal the artwork, or the source code or the text itself, you are probably fine.
That being said, if you are just looking to have a site like Stack Overflow on a particular topic, Stack Exchange is the way to go!