Backup to External in Recovery Mode (Terminal or Time Machine)

Out of the blue, my iMac (late 2010, running El Capitan) started refusing to boot. The progress bar would take forever to reach 100%, and then it would just stop there once it reached it; all the while the fan running loudly. I am able to boot into Recovery Mode, and I can even run Disk Utility (it doesn't detect any problems.) However, when I attempt to reinstall the OS, it tells me: > This disk has S.M.A.R.T. errors. > > This disk has a hardware problem that can't be repaired. Back up as much of the data as possible and replace the disk. I have an outdated Time Machine backup, made before I upgraded from Mountain Lion to El Capitan. Most of the data in the backup is the same, but there are a few GBs of new files I'd like to backup as well. Since Disk Utility tells me everything is okay, am I able to somehow mount an external, and backup files? Possibly through Terminal? And If I were to use the drive with the old Time Machine backup on it, would I have to delete the backup before writing new data to the drive? **I'm not very computer savvy, but I can manage my way through Terminal with a lot of instruction. So please, try to answer in terms I will understand. Thanks.**