Ubuntu boots into emergency mode, home folder empty

After laptop (Lenovo T520) fell on the ground, my HDD (Crucial_CT240M500SSD3) started making clicking sounds during boot and at startup started receiving following message: 2100: Detection error on HDD0 (Main HDD)

So I presume my disk has been damaged and cannot reach login screen in Ubuntu as it always boots into Emergency mode.

You are in emergency mode. 
After logging in, type "journalctl -xb" to view system logs, 
"systemctl reboot" to reboot, "systemctl default" 
or ^D to try again to boot into default mode". 

My main goal is to retreive personal data and save it to an external disk.

The problem is that my /home directory is empty. I have found many methods which point toward resolving the issue, but nothing really worked and I don't know what else to try.

In my /etc/fstab I have separate entries for / and /home, but I haven't created a separate partition for home during Ubuntu installation (at least not manually):

# <file system> <mount point>   <type>  <options>       <dump>  <pass>
# / was on /dev/sdb1 during installation
UUID=99d3a843-b520-41c0-bc80-e64fb67439c8 /               ext4    errors=remount-ro 0       1
# /home was on /dev/sda1 during installation
UUID=f0b370b3-90d8-46ca-956f-f620b5c40536 /home           ext4    defaults        0       2  
# swap was on /dev/sdb5 during installation
UUID=0e4c9122-3317-4478-bc6b-852415505118 none            swap    sw              0       0

journalctl -xb snippet where the mountin of /home UUID fails

...
dec 10 12:11:18 jp-pc kernel: input: HDA NVidia HDMI/DP,pcm=8 as /devices/pci0000:00/0000:00:01.0/0000:01:00.1/sound/card1/input17
dec 10 12:11:18 jp-pc systemd[1]: Received SIGRTMIN+20 from PID 384 (plymouthd).
dec 10 12:11:27 jp-pc kernel: thinkpad_acpi: EC reports that Thermal Table has changed
dec 10 12:12:45 jp-pc systemd[1]: dev-disk-by\x2duuid-f0b370b3\x2d90d8\x2d46ca\x2d956f\x2df620b5c40536.device: Job dev-disk-by\x2duuid-f0b370b3\x2d90d8\x2d46ca\x2d956f\x2df620b5c40536.device/start timed out.
dec 10 12:12:45 jp-pc systemd[1]: Timed out waiting for device dev-disk-by\x2duuid-f0b370b3\x2d90d8\x2d46ca\x2d956f\x2df620b5c40536.device.
-- Subject: Unit dev-disk-by\x2duuid-f0b370b3\x2d90d8\x2d46ca\x2d956f\x2df620b5c40536.device has failed
-- Defined-By: systemd
-- ...
-- 
-- Unit dev-disk-by\x2duuid-f0b370b3\x2d90d8\x2d46ca\x2d956f\x2df620b5c40536.device has failed.
-- 
-- The result is RESULT.
dec 10 12:12:45 jp-pc systemd[1]: Dependency failed for /home.
-- Subject: Unit home.mount has failed
-- Defined-By: systemd
-- ...
-- 
-- Unit home.mount has failed.
-- 
-- The result is RESULT.
dec 10 12:12:45 jp-pc systemd[1]: Dependency failed for Local File Systems.
-- Subject: Unit local-fs.target has failed
-- Defined-By: systemd
-- ..
-- 
-- Unit local-fs.target has failed.
-- 
-- The result is RESULT.
dec 10 12:12:45 jp-pc systemd[1]: Dependency failed for Clean up any mess left by 0dns-up.
-- Subject: Unit dns-clean.service has failed
-- Defined-By: systemd
-- ...
-- 
-- Unit dns-clean.service has failed.
-- 
-- The result is RESULT.
dec 10 12:12:45 jp-pc systemd[1]: dns-clean.service: Job dns-clean.service/start failed with result 'dependency'.
dec 10 12:12:45 jp-pc systemd[1]: local-fs.target: Job local-fs.target/start failed with result 'dependency'.
dec 10 12:12:45 jp-pc systemd[1]: local-fs.target: Triggering OnFailure= dependencies.
dec 10 12:12:45 jp-pc systemd[1]: home.mount: Job home.mount/start failed with result 'dependency'.
dec 10 12:12:45 jp-pc systemd[1]: Dependency failed for File System Check on /dev/disk/by-uuid/f0b370b3-90d8-46ca-956f-f620b5c40536.
-- Subject: Unit systemd-fsck@dev-disk-by\x2duuid-f0b370b3\x2d90d8\x2d46ca\x2d956f\x2df620b5c40536.service has failed
-- Defined-By: systemd
-- ...
-- 
-- Unit systemd-fsck@dev-disk-by\x2duuid-f0b370b3\x2d90d8\x2d46ca\x2d956f\x2df620b5c40536.service has failed.
-- 
-- The result is RESULT.
dec 10 12:12:45 jp-pc systemd[1]: systemd-fsck@dev-disk-by\x2duuid-f0b370b3\x2d90d8\x2d46ca\x2d956f\x2df620b5c40536.service: Job systemd-fsck@dev-disk-by\x2duuid-f0b370b3\x2d90d8\x2d46ca\x2d956f\x2df620b5c40536.service/start failed with result 'dependency'.
dec 10 12:12:45 jp-pc systemd[1]: dev-disk-by\x2duuid-f0b370b3\x2d90d8\x2d46ca\x2d956f\x2df620b5c40536.device: Job dev-disk-by\x2duuid-f0b370b3\x2d90d8\x2d46ca\x2d956f\x2df620b5c40536.device/start failed with result 'timeout'.
...

fdisk says that partition does not start on physical sector boundary (I noticed that sda2 and sda5 reside on the same interval):

$ fdisk -l

Disk /dev/sda: 223.57 GiB, 240057409536 bytes, 468862128 sectors
Disk model: Crucial_CT240M50
Units: sectors of 1 * 512 = 512 bytes
Sector size (logical/physical): 512 bytes / 4096 bytes
I/O size (minimum/optimal): 4096 bytes / 4096 bytes
Disklabel type: dos
Disk identifier: 0x021adb68

Device     Boot     Start       End   Sectors   Size Id Type
/dev/sda1  *         2048 429799423 429797376 204.9G 83 Linux
/dev/sda2       429801470 468860927  39059458  18.6G  5 Extended
/dev/sda5       429801472 468860927  39059456  18.6G 82 Linux swap / Solaris

Partition 2 does not start on physical sector boundary.

blkid (removed loop entries for readability):

$ blkid
/dev/sda1: UUID="99d3a843-b520-41c0-bc80-e64fb67439c8" BLOCK_SIZE="4096" TYPE="ext4" PARTUUID="021adb68-01"
/dev/sda5: UUID="0e4c9122-3317-4478-bc6b-852415505118" TYPE="swap" PARTUUID="021adb68-05"

/dev/sda2 is missing, but when I explicitly call:

$ blkid /dev/sda2
/dev/sda2: PTUUID:"d5620a3d" PTTYPE="dos" PARTUUID="021adb68-02"

lsblk in LiveOS (removed loop entries for readability)

$ lsblk -o NAME,RM,SIZE,TYPE,MOUNTPOINT,UUID
NAME   RM   SIZE RO TYPE MOUNTPOINT   UUID
sda     0 223.6G  0 disk  
├─sda1  0 204.9G  0 part /home/xx/mnt 99d3a843-b520-41c0-bc80-e64fb67439c8
├─sda2  0     1K  0 part  
└─sda5  0  18.6G  0 part              0e4c9122-3317-4478-bc6b-852415505118
sr0     1  1024M  0 rom 

There is no entry for the UUID of /home (taken from /etc/fstab) in /dev/disks/by-uuid/. Only for sda1 and sda5.

Please let me know if any additional info is required.

1. In emergency mode:

$ mount -o rw,remount /
$ mount -a
mount: /home: can't find UUID=f0b370b3-90d8-46ca-956f-f620b5c40536.

2. In LiveOS I ran fsck on /dev/sda1 which found a lot of bad sectors, but now it returns clean.

$ fsck /dev/sda2
fsck.ext2: Attempt to read block from filesystem resulted in short read while trying to open /dev/sda2
Could this be a zero-length partition?

Same error with fsck.ext3 and fsck.ext4

4. Commenting out /home part in /etc/fstab brings me to Ubuntu login screen, but after entering the password it freezes. I presume this is due /home not mounted properly by the system