>I am about to restore my computer to factory settings. Which, I believe, will wipe out drive C:. Drive C: is where all SQL Server 2008 databases are stored. I cannot start SQL Server. So the only way I can "back up" the databases is copy them to an external drive.
>Will I be able to "restore" these .MDF and .LDF file on the SQL Server that I will install later? How?
>TIA.

Are you able to access your SQL Server 2008 on that machine and do a normal backup or detach your databases?

Also, I suggest to install the latest SQL Server afterwards. What is the point of installing outdated software?