Linux boot with broken partition

Question

I have two ext partition on my linux

• One for /
• And one for /home

I wanted to reduce (with this method) the home partition (which was only something like 10% filled) to create a new partition for Windows.

Unfortunately it failed and when I wanted to reboot, linux went in rescue mode, which could have been helpfull unless this rescue mode was not working (It showed two password prompt, and then was working like there was two command prompt at the same time [I had to double every letter to make a command like : ccwwdd //eettcc], so not really doable).

My question is : Can I boot to this linux anyway without going in rescue mode? Like for example disabling this partition (/etc/fstab?).

Also is it possible to repair this partition ? (When I try to mount it with Ext2Fs it says RAW in partition type)

Here is the parted info :

Model: ATA ST9640320AS (scsi)
Disk /dev/sda: 640GB
Sector size (logical/physical): 512B/512B
Partition Table: msdos

Number  Start   End    Size    Type      File system     Flags
 1      1049kB  100GB  100GB   primary   ext4
 2      100GB   420GB  320GB   extended
 5      100GB   110GB  9999MB  logical   linux-swap(v1)
 6      110GB   420GB  310GB   logical
 3      420GB   420GB  367MB   primary   ntfs            boot
 4      420GB   640GB  220GB   primary   ntfs

I think the /home partition is /dev/sda6

Fsck output :

ubuntu@ubuntu:~$ sudo fsck.ext4 /dev/sda6
e2fsck 1.42.9 (4-Feb-2014)
ext2fs_open2: Bad magic number in super-block
fsck.ext4: Superblock invalid, trying backup blocks...
fsck.ext4: Bad magic number in super-block while trying to open /dev/sda6

The superblock could not be read or does not describe a valid ext2/ext3/ext4
filesystem.  If the device is valid and it really contains an ext2/ext3/ext4
filesystem (and not swap or ufs or something else), then the superblock
is corrupt, and you might try running e2fsck with an alternate superblock:
    e2fsck -b 8193 <device>
 or
    e2fsck -b 32768 <device>

Command output :

ubuntu@ubuntu:~$ findmnt;lsblk -f;sudo partx -svb /dev/sda -oNR,START,END,SECTORS,NAME,FLAGS,SCHEMETARGET                           SOURCE     FSTYPE          OPTIONS
/                                /cow       overlayfs       rw,relatime,lowerdir=//filesystem.squashfs,upperdir=/cow
├─/sys                           sysfs      sysfs           rw,nosuid,nodev,noexec,relatime
│ ├─/sys/fs/cgroup                          tmpfs           rw,relatime,size=4k,mode=755
│ │ └─/sys/fs/cgroup/systemd     systemd    cgroup          rw,nosuid,nodev,noexec,relatime,name=systemd
│ ├─/sys/fs/fuse/connections                fusectl         rw,relatime
│ ├─/sys/kernel/debug                       debugfs         rw,relatime
│ ├─/sys/kernel/security                    securityfs      rw,relatime
│ └─/sys/fs/pstore                          pstore          rw,relatime
├─/proc                          proc       proc            rw,nosuid,nodev,noexec,relatime
├─/dev                           udev       devtmpfs        rw,relatime,size=1977152k,nr_inodes=207544,mode=755
│ └─/dev/pts                     devpts     devpts          rw,nosuid,noexec,relatime,gid=5,mode=620,ptmxmode=000
├─/run                           tmpfs      tmpfs           rw,nosuid,noexec,relatime,size=397492k,mode=755
│ ├─/run/lock                               tmpfs           rw,nosuid,nodev,noexec,relatime,size=5120k
│ ├─/run/shm                                tmpfs           rw,nosuid,nodev,relatime
│ └─/run/user                               tmpfs           rw,nosuid,nodev,noexec,relatime,size=102400k,mode=755
│   └─/run/user/999/gvfs         gvfsd-fuse fuse.gvfsd-fuse rw,nosuid,nodev,relatime,user_id=999,group_id=999
├─/cdrom                         /dev/sdb1  vfat            ro,noatime,fmask=0022,dmask=0022,codepage=437,iocharset=iso8859-1,shortname=mixed,errors=remount-ro
├─/rofs                          /dev/loop0 squashfs        ro,noatime
├─/tmp                           tmpfs      tmpfs           rw,nosuid,nodev,relatime
└─/media/ubuntu/04A874F6A874E796 /dev/sda4  fuseblk         rw,nosuid,nodev,relatime,user_id=0,group_id=0,default_permissions,allow_other,blksize=4096
NAME   FSTYPE LABEL MOUNTPOINT
sda                 
├─sda1              
├─sda2              
├─sda3              
├─sda4              /media/ubuntu/04A874F6A874E796
├─sda5              [SWAP]
└─sda6              
sdb                 
└─sdb1              /cdrom
sr0                 
loop0               /rofs
partition: none, disk: /dev/sda, lower: 0, upper: 0
/dev/sda: partition table type 'dos' detected
NR     START        END   SECTORS NAME FLAGS SCHEME
 1      2048  195311615 195309568      0x0   dos
 2 195313662  820317176 625003515      0x0   dos
 3 820318208  821035007    716800      0x80  dos
 4 821035008 1250260991 429225984      0x0   dos
 5 195313664  214843391  19529728      0x0   dos
 6 214843455  820317176 605473722      0x0   dos