How to reinstall a crashed OS X and not lose data?

Question

macOS

1) Macbook stopped loading. Shows when loading a crossed out circle and a folder with a question mark.
2) I tried to reinstall the OS through the Utility, but it does not allow due to the fact that a newer version is installed and the older one cannot be installed.
3) How can I reinstall or restore the system without losing my data (photos, videos, documents)?
Macbook Pro Retina late 2013

Answer 1

I have bad news, it does not see the screw (most likely)
Try to boot from livecd, for example Archlinux dd if=arch.iso of=/dev/sdc bs=4M && sync
and there then try to find something other than a flash drive, and then to rewrite data, read how to mount applefs + and copy the data, or, if you don’t want to, then dd if=/dev/sda of=/dev/sdb (to an external screw or nfs )

Answer 2

What tool did you try to reinstall?
If the disk is alive, then boot into recovery mode, here is a description of all modes. Launch Disk Utility. In it - New image from the folder and specify your folders. But you need an external media where you save this image. And then a reinstall.