luckily, I had this thread in my abo-list, so I was notified about your post!
I'll try to help you ...
0) Connect the external disk, you want to backup your data on ...
1) You have to boot, using a Lion from Recovery Disk, DVD or USB-Stick. You said, you tried permission repair, so I'm shure, you know how to boot from Recovery Disk!
2) If you use FileVault, you have to unlock your HDD. Therefore you have to start your HDD maintaining tool (I've got no idea, how it's called in english, in german it's "Festplatten-Dienstprogramm"
). By right-clicking on the HDD on the left and selecting "activate" or "unlock" you can unlock your disk to "turn off" the FileVault temporarly.
3) Switch to Terminal. I'm shure, there are a LOT of good Terminal-Tutorials, so I wouldn't state pictures or every single step. But if you have ever worked with command line (even in DOS for example), you will be familiar with terminal very soon. Here the short version:
a) change to your /Volume/ directory by using (commands ALWAYS without qoutation marks, of course!): "cd /Volume/"
b) get a list of all aktive Volumes by using "ls -l"
c) open the main HDD by using "cd HDDNAME". IF your HDD is called "MacHDD", you would use "cd MacHDD". btw: Mac is caps-sensitive, so you have to mind the CaPiTaLiZaTiOn ...
d) do the same with every next level of folders. Maybe you just want to copy only your users directory, maybe more. It's a bit like in FINDER now ...
e) after you have found the folder of interest, you just copy it by using this: "cp -r YourFolder /Volumes/ExternalHDD/BackupFolder/". This will copy the folder of interest and everything what's in it to the designated destination (in this case the Folder "BackupFolder" on your external HDD). THIS STEP MAY TAKE A WHILE and you may have to wait very long, until you see the promt again.
f) repeat with every file/folder you want.
TIP: Google for "Copying" or "Moving" of files via terminal on MacOS or something like that. I'm shure, that will bring up enough further tuts and help!
4) Reinstall LION (that's what I did) from scratch and be happy again!
Have a NICE DAY and good luck!!!
PS: When I had the problem, I sent a mail to the support. They answered me and told me, that there is a KernalExtention called com.sierrawireless.driver.SierraDIPSupport, which may have occured the break down. I have never tried to find out, if that was the reason or not. But maybe they will have an idea for your system? Just send them your system.log!