Hello,

Should you have some trouble booting your OSX, as long as there are no hardware defects, you can boot into the Single-User mode to access your files and drives from the command-line. To access this mode hit CMD+S as soon as your here the startup tone and hold it down until the screen turns black and you can see the boot messages. Then put in your external usb drive.

First you have to remount the root partition in write-mode
# mount -uw /

then create a folder where you want to mount the drive
# mkdir /Volumes/USB

and mount the drive
# mount_msdos /dev/disk1s1 /Volumes/USBDrive

Now you can start to copy your files over.

One last note:
the "mount_msdos" command is used if it's a FAT(32) filesystem, there are other commands for other filesystems (like EXFAT etc) and don't forget to replace the "disk1s1" with your drive's partition.