Technically there's no reason why you can't do this. It's not without it's challenges though. Maybe copying the content into wiki pages would be more practical?
However, assuming you do want to go the HTML route, in SharePoint you can create a document library and upload any type of content, including HTML pages. You can also add folders and those will work in the same way as folders in the file system. However, by default SharePoint will prevent the HTML files from being displayed directly in the browser for security reasons (a user could easily modify the HTML files to contain some nefarious script). You can workaround this by checking out this post:
From the post:
You can modify SharePoint’s behavior by changing the Browser File Handling option in the Web Application General Settings of SharePoint 2010. Your options are permissive and strict, with strict being the default.
The difficult part will be to keep the file structure the same so that links etc. still work. For example, if you create a document library called 'WebPages' and upload your content there, all of your html pages will be at http://yoursite.com/webpages/yourpage.html which probably isn't what you want if they were previously at the root. To address this you'd need to use redirects. Rewrites would be nicer but aren't officially supported in any practical capacity since the platform makes extensive use of them internally.
The official statement on support of rewrites can be found here
Using SharePoint Designer, you can change the default homepage to pretty much anything, including one of your uploaded HTML pages. Maybe that's enough if your site is simple and you don't mind breaking existing deep links.
To do that you'd open SharePoint Designer, select All Files from the left bar and then navigate to your document library. Once you find the file you want as the homepage simply right-click on it and select 'Set as Homepage'.