If the database was not affected by the file system problem then you should still be able to connect to it using the same connection string. All of the site content is in the database except images and files uploaded.
To restore a database on another machine you would first have to get a backup of the database by creating a backup using SQL Server Management studio on one machine then restore the backup on a new database on another machine also using SQL Server Management Studio.
Else if you get the database files and copy them to your local machine it "might" be possible to attach to it as an existing database rather than restoring it from a backup, but you'll have to do some googling to learn about that possibility.
Hope that helps,