Downvoting the answer is the first step if you doubt the claims or think they need a citation. You should also leave a comment saying what aspects of the answer you think are controversial and require sourcing. Hopefully that user will provide an acceptable source. If they can not, more downvotes will accumulate. If the user continues this pattern they will eventually face various impediments to posting more answers, including warnings, rate-limiting, and eventually an outright answer ban. By far the most important thing is establishing a community consensus that unsourced claims are not acceptable when a source is called for, so we need to make sure such posters are called out on their lack of sources.
At the moment, that's really all we can do. Once we have pro-temp moderators, there will be more options. Post notices are one such option (as on Skeptics SE). It may be that the default set of post notices includes the "This answer doesn't cite sources" one, but if not, the moderators would have to ask the community managers. Moderators will also be able to warn users who continue such patterns, and suspend users who keep doing this repeatedly after being warned.
As for using obscure sources, that seems like a separate issue. I don't know exactly the right course of action. I know I've already cited obscure books and papers before. Sometimes I was able to find a free link (especially for public domain works), but other times, they were paywalled and I only had access through my university library. Generally speaking, I'd say we should be more concerned with the content than the accessibility. If there's really a doubt over what the source says and it's behind a paywall, ask the poster to quote the relevant sections, but I suspect a lot of users here will have academic affiliations (either in science/math or in history) and so will have access in some way to many of the obscure things cited here.