If you want a program that has an interface very similar to Family Origins, try RootsMagic, a Bruce Buzbee product. Bruce was the author of Family Origins. Version 3 of RootsMagic would import directly from a Family Origins backup (*.ZIP) as well as the data files (*.FOW). I do not think the current Version 4 of RootsMagic will import from the backup but will import from the data files (*.FOW). It will also import a GEDCOM (*.GED) made by any genealogy program. This would be the simplest and safest way.
I am assuming your old machine is functional and you are able to make, or have made, a current GEDCOM from Family Origins and have a method to transfer the GEDCOM file to the new machine.