serve "normal" HTML pages from a Sharepoint server
Solution 1:
You can upload your static HTML into any document library and it will work. Make sure you use relative URLs throughout to save yourself a lot of pain. CSS and images will display however you may have security issues with JavaScript.
Using SharePoint to store this information probably isn't the best idea unless you want to use its version control (or some other SharePoint-specific) functionality. Make sure there is a good business case or it will be much simpler to just host it as alpha1 has suggested.
Solution 2:
You need to set your Browser file handling to "permissive"
Solution 3:
I'm trying this out right now, in a similar situation. I have a Bootstrap-based site, designed and perfectly functional outside of SharePoint, and my task is to stand it up inside of SharePoint, since that's the only server available. I thought I'd have to redevelop the entire thing. But what I've done (so far, just in the testing stages) is use SharePoint's Site Content and Structure settings to replicate the directory structure of my site, and then I've uploaded my files and used the site settings to change the start page of the site from default.aspx to my index.html. And, in my scenario, and much to my surprise, it's working just fine.
To be fair, this particular site doesn't need any of SharePoint's functionality, other than access control. And there may be issues I haven't accounted for yet. But I was shocked to see that it actually worked.
This is a SharePoint Publishing site, BTW. Not sure how it would work (or if it would) in other types of SharePoint sites.