Bingo...
you will need to go to
www.versiontracker.com and download SMART Utility.app
This will show you what the SMART pass/fail in Disk Utility does not.
Like whether there are bad sectors or errors on the drive signaling its impending death.
Even if it is bad, and out of warranty. You likely can spend $50 and DIY or upgrade the size of the drive for a bit more.
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Why do we say HDD problem over DVD dying, is because an OS upgrade off DVD will tend to stress the drive. And if it is failing, it'll start having errors when the mechanism gets hot and eventually this gets bad enough the OS update is aborted.