Restart your Mac and hold down the Alt/Option key. When your machine gets to the point where it looks for an operating system to start from, it will give you the option to choose any suitable drive to start from.
This puts you in a pretty good position to proceed, as you will no longer be using the faulty disk, and you won't be in danger of overwriting any lost data. You can try running Disk Utilities at this point. But it's more likely that the problem will warrant using a data recovery App to retrieve any data you can.
How to delete corrupted files on external Mac drive
Having an external drive is useful for backups and having the ability to install OS X separately, however sometimes these very drives are the problem-child.
There might be corrupt files within the drive, or you might have a completely corrupt external drive, in either case, we presume your primary hard disk on your Mac is operational, meaning you'll be able to launch OS X and run Disk Utilities. It might seem like the obvious choice to go to, but it's by far the best method to solve corrupt files on a hard disk that wants to run alongside OS X.
From Disk Utilities, we suggest running First Aid. If that still doesn't help, you can manually go into the Finder and delete the individual files you think are causing you problems.
If on the other hand, the whole drive is corrupt and you've tried every last resort to extract data from it and failed, then you last-resort option might be to fully format the drive and hope you can at least use it after the deletion process has passed. Before proceeding, we would urge you to try using data recovery software or even visiting a computer specialist to extract the data off your drive.
We have an extended guide on how to format a storage drive for Mac - follow those steps and you should hopefully have a functioning drive.
Use data recovery software on Mac
If you have managed to start your machine from either the Recovery HD, or an external drive, the volume you are trying to recover may not show up. Data recovery software can still work with that drive, so long as the OS can still talk to it. If the data is still on the drive, you still have a reasonable chance of recovering it.
There is one exception though, and that is if you were using an SSD. Because if the nature of SSDs, it is often impossible to retrieve data from them, no matter which tool you are using. This is because, unlike HDDs, SSDs would be very slow indeed if they had to save data to a block, or cell, that already has data on it. HDDs can merrily over write data over and over again. But SSDs must first erase that data to be able to quickly record new data. So SSDs have various strategies to manage this. OS X uses the TRIM system, which zeroes out the cells holding data is deleted, as part of is disk management. Other SSDs will always use a similar system, because if they needed erase and record at the same time, they would end up being slower than HDDs.
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