Upgrading from WordPress 3.8.2 to latest – only files and SQL dump available

If you can get an older version to work with the database (set up the config file to point to your local version of the DB), then you can try exporting it (via Export tool), and importing into another instance of the latest WP install (install a base site with a new DB).

If only the content is important to you, you could extract just the posts table to get the post/page content via phpMyAdmin. Then import that into the new instance either manually or with some PHP programming (create a loop that goes through each of the post/page content of the old DB, then use WP functions to create a new post).

You should be able to find guidance on the above processes with a bit of searching.