The right adapter might let you recover your files, but there’s no guarantee. A lot of it depends on the physical condition of the drive and the way your files are stored on it.
But first, a reprimand. If you backed up your files daily, you wouldn’t have this problem. You could just restore the files from your backup. It really isn’t difficult.
Okay, let’s get on with your problem.
A SATA/USB adapter temporarily turns an internal drive into an external one. You remove the drive from the computer, plug the SATA end of the adapter into your drive, then plug the USB end into another computer. Windows—or whatever operating system the computer uses—should treat it like a flash drive or an external hard drive, allowing you to access the files and copy them elsewhere.
Unless your drive is the part of your PC that died. (You can test this beforehand by booting a live version of Linux and seeing if you can access your files.) If your files remain inaccessible, you’ll have to send the drive to a data recovery service.
Another possible problem: If you use something stronger than a Home edition of Windows, you might also run into a problem with Windows’ own built-in encryption. The Encrypting File System (EFS, which encrypts selected files and folders), and Bitlocker (which encryptions whole partitions and drives), require a special key to decrypt on a computer other than your own. If you didn’t save that key or don’t remember where you put it, your files are gone for good.
That’s why Microsoft doesn’t make these tools available in the Home editions. It’s also why I recommend using a third-party password-based encryption tool such as Veracrypt.
The good news is that if the drive is in good shape, and you haven’t used Microsoft encryption, you will probably be able to recover your files.
One more suggestion: Instead of buying a SATA/USB adapter, buy a SATA/USB enclosure. An enclosure is basically an adapter in an enclosed box, turning the former internal drive into a full-fledged external one.
Then you can use that external drive for backup. That way, you won’t have this problem again.