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I have an outdated Time Machine backup, made before I upgraded from Mountain Lion to El Capitan.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.
Whatever you do, leave the faulty disk alone until you know what you are going to do. The risk that the disk breaks entirely any moment now is very high. 

This answer assumes you don't have FileVault enabled.


Get an external backup disk that has enough capacity and connect it.
Press cmd+alt+R before the boot screen appears to enter internet recovery mode. This loads the recovery mode from the internet, which will be very slow. The advantage is that there is no stress on the faulty disk.


Method A – If you want to save individual files:


Go to the menu bar and select Terminal.
Copy the files you want to rescue using cp. For instance, if you have a folder rescueme on your desktop, you can use the following command: 

cp -r /Volumes/mydisk/Users/myusername/Desktop/rescueme /Volumes/mybackupdisk/
This will copy all files and subfolders of rescueme onto your backup drive called mybackupdisk.


Method B – If you want to make a full backup:


Enter the Disk Utility tool
Select the Disk that you want to back up
In the menu bar select File &gt; New Image &gt; Image from “mydisk”
Select your options. E.g. you may want to compress the image to save some space on the backup disk.
Press Save and wait.


It is likely that some parts of your disk are not readable anymore. Therefore the method B may not work and you should first try to save the most important files manually (i.e. method A).

Method C - if you have another Mac:


Connect the two computers, e.g. with a Thunderbolt cable.
On your broken iMac, press T at the boot screen to enter target disk mode.
Now you can use the Finder to browse the broken disk and copy files.