Noton Ghost 14 says my hard disk is about to fail and won't do a backup

you can also try to repair the sectors with HDD regenerator - it does not touch the data !
http://www.dposoft.net/      

I ran CHKDSK and it said there were 8 errors. In the Event Log there were many errors, all about the hard drive. I found my hard drive is a ST380021A (9P519 Hard Drive, 80G, I, Half Height 7.2K, 40G/P, SEAGATE SNOWMASS... ) and downloaded both the Windows and DOS version of their SeaTools software. Running the Windows version's tests resulted in 1. Short DST - FAIL 2. Long DST - FAIL 3. Short Generic - Pass. I didn't run the long Generic since they recommended burning a bootable cd with the DOS SeaTools and running it at that level. Upon boot, it ran and I ran the Long DST and it found and repaired three errors. Unfortunately, after that I rebooted and ran the Windows SeaTools Short DST test and it once again failed.
I will now try out the HDD regenerator as suggested by nobus.
In the meantime, I want to go ahead and get a drive to replace the troubled one, so do I need to keep with the same size hard drive in order for the Ghost Restore to work properly or can I upgrade a bit in the process? I was looking at a Western Digital Caviar SE Hard Drive - 160GB - 7200rpm - Ultra ATA/100 (ATA-6) - IDE/EIDE - Internal (WD1600AAJB) for $55.

You can replace it with any IDE drive.    The only issue is whether or not your system has 48-bit logical block addressing support.    If not, then any drive larger than 137GB (128GB in "computer-ese") will be limited to the 28-bit LBA limit (137/128GB) -- it will still work, it just won't be seen as any larger than that, regardless of its actual size.    

I've ordered a new hard drive and will be replacing the failing old one. Looking at the HDD Regenerator info, however, it says:
'Since the program does not change the logical structure of a hard drive, the file system may still show some sectors marked earlier as "bad", and other disk utilities such as Scandisk will detect logical bad sectors even though the disk has been successfully regenerated and is no longer damaged by physical bad sectors. If you want to remove these marks, repartition the hard disk drive.'
Would Norton Ghost still refuse to back up the drive since it's still marked as "bad"? I was thinking that if it was repaired, I'd make a good backup to use to restore to the newly installed hard drive. I won't bother with HDD Regenerator or the additional backup if Ghost won't honor the repaired sectors. Or is it just disk utilites that see it that way and other programs just try (and succeed) anyway.

As soon as you complete the migration run on new HDD CHKDSK x:/r where X: is the drive letter of your partitions. Since then no bad sector will be found.

Hard Disk Regenerator uses a statistical analysis of the "bad" sectors to rewrite them with the "most likely" data they contain.    Basically it simply reads them MANY times and compares the results for each bit, determining which bit is most likely the correct value ... and then confirming that the sector's ECC is correct with the bits it has assigned.    It uses a statistical technique to retry if the ECC is incorrect until it finds the right combination ... or simply determines it can't fix the sector.     This is very similar to the "DynaStat" technique that Spinrite uses.

For those sectors it is able to successfully regenerate, it reallocates them to another sector, and then creates a hidden file that occupies those sectors that were previously marked as bad -- this prevents them from being used again without invoking the drive's remapping feature.    Spinrite would either remap them, or -- if it determined they were fully recovered -- simply allow them to be used.   (Thus eliminating the "still bad" marking issue)

In any event, you can make an image with Ghost at that point -- you may simply have to use the  "ignore bad sectors" option ... but since you won't have any significant data in those sectors, it'll be fine.    And they won't be marked as bad on the new disk.

Nevertheless, a clean install is always best -- and especially since you don't really know what may have been corrupted/lost with the issues you've had with the old drive.    If you carefully save all of your data to another drive (an external one is fine) -- don't forget e-mail, address book, favorites, etc. that aren't kept in "My Documents" -- then all you have to do is reload the OS, load all your programs, and then restore all of your data.     If you have programs you don't have the install media for, there are ways to transfer those to a new OS load as well -- just post back if that's an issue.

Ok. I've got my work cut out for me. I'll have to wait until I get the new hard drive in a few days (we're ultra-rural) and then I'll get crackin'. Thanks for all the input. I'll be back and let you know how it goes.

just a question : did you run HDD regenerator, and what were the results if you ran it ?

Weird! I just ran HDD Regenerator and it found NO ERRORS at all. What's that about?

that only means that the surface and the signals are ok
i would run a disk diag to be sure about it's status :  http://www.tacktech.com/display.cfm?ttid=287

Ok. I'll try that. I am at work and the troubled drive is at home so I'll get to it as soon as I can.

I'm back (very quickly). As it turns out that last link sent me around the horn to the same place I went a few days ago to get the SeaTools, which I ran with the results previously reported above:
"Running the Windows version's tests resulted in 1. Short DST - FAIL 2. Long DST - FAIL 3. Short Generic - Pass. I didn't run the long Generic since they recommended burning a bootable cd with the DOS SeaTools and running it at that level. Upon boot, it ran and I ran the Long DST and it found and repaired three errors. Unfortunately, after that I rebooted and ran the Windows SeaTools Short DST test and it once again failed."
So it would seem I just need to patiently await the arrival of the hard drive I ordered.

Yes, I'd just wait for the new drive and get it installed.   Once hard drives start exhibiting issues, it's not worth trying to nurse them into the last few weeks/months of life.    They're simply not reliable at that stage.    If you want to keep using it for auxiliary storage, just use an external USB bridge; reformat the drive; and use it to store miscellaneous backups or unimportant stuff (downloads, etc.).    But in general the best thing to do is just trash it.

(I have a bunch of older, low-capacity drives I use for a variety of things -- but I toss any drive that isn't error-free with a Level-2 pass with Spinrite)

Thanks for the guidance.