Do not power the computer off unless you are hearing clicks or strange sounds.
Get a external drive - buy one or borrow one - just get one and start with your most valuable data and work back!
Usually, you will have time to safely remove data from the hard drive before it dies on you.
Albeit, this isn't always the case unfortunately.
Having data extracted from a hard drive after it has physically failed is an extremely expensive ordeal which most people can't afford.
Do NOT try to take a hard drive apart under any circumstances if you are not in a Class 100 Clean Room. This is where hard drives are dismantled for repair and if you value your data, or it is worth having this work done, it should be done.
Once you open a hard drive in normal conditions, you've ruined it due to dust.
If the problem is NOT physical, the repair is much less expensive and some of it, if you have some knowledge of hard drives and how they work logically, there is a wide variety of software, some free, that will help you get the hard drive back up and running.
Sometimes the partition tables, file structure, directory structure or system files on the drive become corrupted and if you know what you're doing, you can fix this - or take it to someone who will do it for you.
If you DON'T know what you are doing, you could irreversibly destroy your data!
Final Thoughts...
It's a good time for me to review this area also. i work as a songwriter & music producer and my operating basis for recording is to keep all audio & production files on an external drive which works more efficiently with the production software.. so in this situation i also need a back up of my external drive..
On a kinda related topic i just implemented the use of the WordPress plugin "UpdraftPlus Backup" to automatically do a full site backup to my Google Drive once a week.. it does make you sleep a little easier at night :~)