Disk Utility says "drive can't be repaired, disk and restore your backed up files."

My Mac Book Pro froze yesterday and I had to force restart it. It had been showing me messages that my hard drive disk was full for a while and I was in the middle of transferring files to my external hard drive and I accidentally put a huge large file from my external hard drive into the trash on my laptop which makes me wonder if that means I added more GB to a computer that was already very full. Anyway, after freezing I restarted it and the desktop wouldn't load properly. The desktop picture was there, but folders on desktop did not appear, finder bar was missing and the dock was acting super slow and nothing worked when clicked on. Then the next few attempts were around the same result and eventually became stuck on the apple logo, spinning gear and a progress bar that was not loading at all.

Here's what I've tried so far:

Safe mode- nothing happens

Command + R - I am unsure how to work this since I can't seem to find an internet connection when it prompts for one. None of the internet connections have much signal and my own information doesn't seem to load since the gear spins to find the internet connection name and password I typed to load it but can't seem to find it. Perhaps I have the wrong internet information, but I don't think so.

I tried Command + S to use in single user mode and type in a command to help fix the drive, BUT, even though the black screen and white lettering mode did show up, it remained in a place where I was not allowed to type anything and the last thing written at the bottom of the screen was something about set up for bluetooth, I think. I tried this twice and both times it stopped in the same place and I could not type a command.

Disk Utility using installation dvd to verify and repair disk. Verify says disk needs repair and then repair has some red colored phrases: Invalid volume file count. (It should be 786374 instead of 786274) Incorrect number of thread records. And there was another problem listed but I do not remember it. Then, near the end of repairing it says: "Disk Utility stopped repairing Macintosh HD." It tells me it couldn't repair the disk and that I should back up as many of my files as possible, reformat the disk and restore backed up files.

That would be fine and dandy, but I have no idea how to back up my files and do all this. I have no idea how to exactly get the data off my hard drive. If someone does, please list a step by step guide for me on how to do so.

I have an external hard drive and I tried plugging in and selecting the hard drive and making a "New Image" to copy to the external hard drive destination, but it just stops a few seconds into it and tells me there is an input/output error. I tried read only, read/write only, compressed options for making a copy and all yield the same result.

In Disk Utility, I believe I have a partition -the Macintosh HD. I have listed on the left hand side: 320.07 GB Hitachi HTSS45.....etc. and below it in non-bolded, grey lettering it says "Macintosh HD."

The Hitachi, when clicked upon, shows the S.M.A.R.T. status as Verified and the Macintosh HD doesn't show much in details except for "Not mounted."

At one point, the Macintosh HD was in bold lettering and seemed to be mounted for a few minutes. When that happened, I was able to see the details reading: Mount Point: /Volumes/Macintosh HD, Capacity: 319.73 GB, Available: 11.92 GB, Used: 307.81 GB, etc.

I'm getting nervous to keep turning on and off the computer on force start and shut downs, especially since I have been reading up that the longer the computer is on, the more the hard drive is in use and the more chance for it to start deleting or overwriting my files. Is this true?

It's not mounted anymore since it was momentarily mounted. I just want to get my files off the disk and save them to my external hard drive. That is my main goal right now. What can I do?

Thank you in advance for your help!