No, you won't lose any files, just by booting to your recovery system.
If Disk Utility reports a problem that needs a repair of the file system, it generally cannot fix a directory problem WHILE you are booted to the same system. The next step is to boot to ANOTHER system, and repair your hard drive from there. You have much greater chance for success then...
The Repair system is on a hidden partition on your hard drive.
Very simple to use... Restart your Mac, while holding Command-R.
You will boot to the repair system, with a window showing various choices. Choose Disk Utility. Now, you will be able to choose Repair Disk (not Repair Disk Permissions, which will not help in this situation)
If the Repair Disk reports errors that it fixes, run Repair Disk again, until no problems are found. If the same problem is listed every time, then the Disk Utility may not be capable of the repair, and you may need to go to something more capable, such as Disk Warrior (you would need to purchase that)
If the Repair Disk completes successfully, you can restart your Mac. You may then find that the backup that you were trying to perform - may finish this time.
Come back if you get errors that you can't fix with Disk Utility.