If I answer a question making various historical claims, how strict should we as a community be about what kinds of references I use to back up my claims?
No references at all is to be discouraged and downvoted.
Credible historical books and articles (for some value of "credible") are acceptable, and you should cite the specific book/article and page number. In short: just tell us where you read what you're saying. Especially if it's an off-line or otherwise somewhat inaccessible source, please also mention any information about where that author claims to have gotten their facts.
Anything better (such as original sources) is obviously ideal, but it seems like it would be a bit stifling to require original sources all the time.