Well chances are you will lose your data if you reinstall as the installer will format the partitions. It is possible to reinstall and not format the partition is your data provided it is currently partitioned to have your file on their own partition. I am going to assume that the drive is 1 big partition and that this is not an option.
In the event that you have your drive partitioned as 1 big partition you best option is booting from a LiveCD. I have not used Fedora since Fedora Core 3 so I am way out of date. I don't know if Fedora comes on a LiveCD or not. If it doesn't just download a copy of the Ubuntu install cd, I know it is a LiveCD.
1.) Boot from the Live CD.
2.) Create a folder to mount your hard drive to
3.) Mount your hard drive to the newly created folder. (Your hard drive will not automatically be mounted by the LiveCD unless you run an installer. You will have to manually do this. Depending on the type of hard drive the identifier may be different. Now days most home build computers use SATA hard drives which are designated as /dev/sda, /dev/sdb, etc. If you only have 1 hard drive and it is 1 partition then you would mount /dev/sda1.
mount /dev/sda1 /path/to/newly/created/folder
You may also need to use the -t option and specify the file system type. I do not know what you are currently using but default I think when you install is either ext3 or ext4.
Once you have your drive mounted to the folder you can browse the folder to view the hard drive.
Another option is to run fsck on your hard drive to repair any errors on it.
**you will need root access to do all of this. I am not sure if this is given by default in a Fedora LiveCD, but in ubuntu you will put sudo before all commands. example: sudo fsck /dev/sda1