It's not the best option, but if there's a final critical file you need to get, this move won't damage the hard drive any further and it may pull off the miracle you need in a pinch.
Back up the Users folder first
The Users folder is where most of your critical stuff lives; and if you lose it, your day is going to be that much worse. Locate your Users folder (Hard Drive > Users) and then slowly copy data from it, one file or one small folder at a time.
Remember that you're dealing with a hard drive that has absorbed a powerful physical blow--an electronic component's equivalent of being on the receiving end of a haymaker from the heavyweight champ when all it wanted to do was step out for a bagel. Take it easy on your shell-shocked drive.
The pricey-but-worth-it approach
Alsoft's DiskWarrior is a great application for taking a damaged hard drive and rehabbing it to a point at which you can recover its data. At $100, the app is pricey, but the investment pays off; I've rebuilt more hard drives with DiskWarrior than I care to mention.
Prosoft Engineering's Data Rescue 3 lets you mount and work with drives that may be too damaged to mount under OS X. It can also help you work around damaged sectors on the drive to recover every savable chunk of data. At $99, it's another fairly hefty investment; but between these two apps, you can rebuild and recover an amazing amount of data, if the drive's mechanical functions are still operating normally after the drop.
No matter what, back it up
Back up your data to multiple destinations--to iCloud, to the $6 external flash drive you bought at the pharmacy, to Carbonite, to Backblaze, to an external hard drive via Time Machine, and elsewhere.
Hard-drive storage is cheap and backup services are affordable. Going with any of these options can save a whole lot of misery down the road, especially if the alternative turns out to involve a four-figure estimate from a drive recovery service.
The SSD option
With a solid-state drive, you pay a premium for a blazing-fast drive that doesn't have a ton of storage space on it. But beyond its superior speed, an SSD has one distinct advantage over a traditional hard drive: It contains no moving parts and is therefore essentially shock-proof.
An SSD isn't invincible, but it can operate under a wider range of temperatures and can endure thumps and lumps more readily than a hard drive equipped with moving parts and spinning platters.
If possible, look into upgrading your Mac with a SSD for its main drive. If the sum of your data won't fit on that drive, look into putting your operating system and your critical data on that drive and your other data on either a second internal hard drive or an external hard drive.
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