Is it my system or computer / HDD broke? What can I do about this? (Linux Mint)

I've been using Linux Mint casually for a few months. Suddenly yesterday when I enter my HDD password I get the following input:

BusyBox v.1.22.1 (Ubuntu 1:1.22.0-15 ubuntu1) built-in shell (ash) Enter 'help' for a list of built-inn comands (initramfs)_

This happened when using the standard boot option which is "Linux Mint 18.3 Cinnamon 64-bit". Using help function gives me a long list of commands, none of which are comprehensible to me since I'm a complete newb who just wanted to use Linux for regular office stuff without worrying too much about advanced techniques.

So I started using other boot options under a name "Advanced options for Linux Mint 18.3 Cinnamon 64 bit". Choosing this option gives me 3 new boot options:
- Linux Mint 18.3 Cinnamon 64-bit using Linux 4.10.0-38-generic
- Linux Mint 18.3 Cinnamon 64-bit with Linux 4.10.0-38-generic (upstart)
- Linux Mint 18.3 Cinnamon 64-bit with Linux 4.10.0-38-generic (recovery mode)

The first two of these options take me to the same screen as described in quoted text. The third option does so too but gives me a long output of what's happening. Unfortunately I was unable to capture it all and I can't scroll it back (if it's possible and necessary then I would be grateful for explaining to me how to do so). I have written down the last and I suppose crucial part of the output which is as follows.

[ 34.030720] raid6: .... xor() 1856 MB/s, rmw enabled [ 34.030763] raid6: using ssse3x2 recovery algorithm [ 34.033969] xor: automatically using best checksumming function avx [ 34.075178] Btrfs loaded, crc32c=crc 32c-intel Scanning for Btrfs filesystem done. Begin: Will now check root file system ... fsck from util-linux 2.27.1 [/sbin/fsck.ext4 (1) -- /der/mapper/mint--vg-root] fsck.ext4 -a -co /dev/mapper/mint--vg-root /dev/mapper/mint--vg-root contains a file system with errors, check forced /dev/mapper/mint--vg-root: Inodes that were part of a corrupted orhan linked list found /dev/mapper/mint--vg-root: UNEXPECTED INCONSISTENCY; RUN fsck MANUALLY (i.e. without -a or -p options) fsck exited with status code 4 done. Failure: File system check of the root file system failed The root filesystem on /dev/mapper/mint--vg-root requires a manual fsck

Moreover I have two more booting options which take me to some kind of an application which I don't understand really but it appears to be some scanning app. I will experiment with it now to see if I can get some info from there but first I wanted to post it already in case the data provided is already sufficent. I will update my post with eventual findings (or lack of findings) in upcoming hours. These two additional booting options are:
- Memory test (memtest86+)
- Memory test (memtest86+, serial console 115200)

Now my questions is - does anyone know what is going on here? Is it a problem with system or rather with my computer as a whole (f.e. HDD)? I really need this laptop and I'm fine with reinstalling Linux because I have backups of the most important files (passwords). But if there's some fix I'd rather use it. If it's HDD thing then I'm screwed. Someone please help me.

If you need any more data then feel free to ask. Thank you in advance!

yes, your filesystem is now corrupted. Boot into recovery mode and run fsck manually.
From this information I cannot tell you if your HDD is ok or not, but hopefully that is working. fsck will tell us more.

I strongly recommend to have a second internet-connected machine handy and do web research while you attempt to fix this, e.g.:
"linux how to run fsck manually" etc.

What about just downloading my distro again and installing it all over? I don't seem to have anything important on my computer that isn't backed up. Do you think it should work? Maybe it's a "stupid" solution but I don't really have anything to fight for and I need my computer ASAP.

no, it is not a stupid question, you can do it - obviously it depends on you. Probably faster. But remember, if your hardware has been damaged it won't work too.

Yes, it's your hard drive which is the problem. Either the hardware is bad or (more likely) you just have a corruption in the filesystem, which can easily be corrected.

To explain a bit what you observed (which may not interest you but I think newbies always deserve an explanation, otherwise they think Linux is just gobbledegook):

Most Linux distros use a stock or generic kernel which does not contain drivers for many kinds of hardware. These drivers are on disk, ready to be loaded on demand. But how can the kernel load them from disk if it doesn't have suitable drivers for reading the disk? The answer is that the bootloader loads a miniature filesystem called an initramfs or initrd into memory and the kernel treats that as its first root filesystem. Then it can load from the initramfs the drivers it needs to access the hard drive. If after that, it still can't access the hard drive, it tells you, and this was the first message you got. The prompt "(initramfs)_" tells you that you haven't succeeded in mounting a hard drive partition yet, and the other references are to the software that initramfs uses to save space: ash and busybox. Ash is a very small command processor or shell and busybox is a collection of useful commands stuffed into a single program.

The initramfs also contains a separate fsck program for checking filesystems, so you can run it from there. The more verbose recovery mode boot tells you which device you must use fsck on.