Data Backup: Physically transferring your drive
If you are not familiar working inside a computer, then DO NOT ATTEMPT this method. If either system you are planning on using has any kind of warranty, you will most likely void that warranty by trying to remove that drive. Check with your system manufacturer to find out their exact policies on this kind of thing. In addition, if you are not extremely careful during this process, you could damage different components in your system.
As this method should really only be tried by those with a pretty good understanding of the workings of a computer and how to transfer a drive, we'll only be giving basic instruction here.
How this works is that if you have a second system that is still up and running, you can pull the drive you are wanting to use, and you can install it temporarily into the second system to access it like a second drive in the system. You will need to make sure that both are SATA (Or IDE if they are older) to make sure you'll be able to do this.
Once you transfer the drive over, there are a few things you'll want to keep in mind. First, when you turn the system on it may try to boot to the drive where Windows is broken. If it does that you'll want to bring up the temporary boot menu (Look at Steps 1 and 2 on the "Bootable Backup" method below to see how to do that) and select the proper drive for that system.
Once you get into your operating system if it doesn't show up at all you may have to assign a drive letter to it. On windows 8 or 10 right click on the start menu and go to "Disk Management". On windows 7 you should be able to search for "Disk Management". Once you are there you should be able to right click on the drive and hit "Change drive letter and paths." At that point give it a letter and it should show up.
From there you should be able to access it much like a flashdrive or secondary drive in the system to pull off any needed data.
Once you've backed up your data, you'll most likely want to reinstall Windows itself. Click this guide to get started
This is a simpler method of backing up data requiring only a Windows bootable.
If the method you just tried didn't work for you, then you'll want to try creating a Linux flashdrive. This method will let you grab more than one file at a time.