It's good advice when people are told to back up their data in advance. However, since it's human nature to ignore even the most critical things, the repercussions of a failed hard drive often pushes the subject to seek desperate measures at whatever cost.
Recovery of backed up data from damaged hard drive
Switching your system off and back on again
Switching the system off and on again is something that has quickly become a cliche. However, it's still sound advice because it normally solves minor glitches in the system. For this reason, switching off/on is the first bit of advice that people will be told to follow. But unfortunately, if one is dealing with a dying hard drive, it may not really survive the experience, and this may lead to a stickier situation. Even if a system has an optical disk drives, one may not have the option of booting from another disk.
But if the problem of the hard drive isn't too terminal, restarting may show Recovery HD (if one is using a Mac computer). Recovery HD is a special partition in the system that will not show up until it is most needed. Also, it has been shown that booting from this partition doesn't affect the main hard drive in any way, so there should be no worries of overwriting the data on the hard drive.
The use of data recovery software
When systems fail, victims always rush to get a data recovery software to salvage the lost data. And the worst thing about this software is that you'll only use it once when you have encountered a failing hard drive. In fact, the developers of these programs have you at their mercy. When you approach them because you have lost crucial data, they can charge you whatever price they want to. Good thing, however, is that this is a highly competitive industry, and for this reason, the pricing structure has somewhat evolved to achieve an equilibrium state. Most of them will cost $99, although some developers may charge more depending on the features they offer.
Another good news is that developers don't always want their consumers to spend the $99, then come back today, tomorrow and the day after tomorrow with complains that the thing didn't work. For this reason, all of them will have a trial version that scans the problematic disk to highlight what can be saved. From there, you can decide if all those fragments of lost data is worth the investment.
SSDs vs HDDs
The nature of SSDs don't allow people to recover data from them no matter which tools are used. SSDs would be very very slow if they were subject to the task of saving data to a cell or block that has another data on it.
On the other hand, HDDs can over write any type of data repeatedly -- different from SSDs which must first erase fully to record new information.
Recovery of backed up data from damaged hard drive is possible in many cases. But if the problem is severe, one can seek the services of a professional to help them save the data.