You are using a USB install and you have some problem with the USB drive that you have installed on as it is reporting a lot of errors:
Jun 29 10:48:05 osmc kernel: EXT4-fs error (device sda1): ext4_iget:4056: inode #859: comm chown: bad extended attribute block 1073741824
Jun 29 10:48:05 osmc kernel: EXT4-fs error (device sda1): ext4_ext_check_inode:495: inode #857: comm chown: pblk 0 bad header/extent: invalid eh_entries - magic f30a, entries 65, max 4(4), depth 0(0)
Jun 29 10:48:05 osmc kernel: EXT4-fs error (device sda1): ext4_ext_check_inode:495: inode #860: comm chown: pblk 0 bad header/extent: invalid extent entries - magic f30a, entries 1, max 4(4), depth 0(0)
Jun 29 10:48:48 osmc kernel: EXT4-fs error (device sda1): ext4_ext_check_inode:495: inode #857: comm dpkg: pblk 0 bad header/extent: invalid eh_entries - magic f30a, entries 65, max 4(4), depth 0(0)
Jun 29 10:48:49 osmc kernel: EXT4-fs error (device sda1): ext4_iget:4056: inode #859: comm chown: bad extended attribute block 1073741824
Jun 29 10:48:49 osmc kernel: EXT4-fs error (device sda1): ext4_ext_check_inode:495: inode #857: comm chown: pblk 0 bad header/extent: invalid eh_entries - magic f30a, entries 65, max 4(4), depth 0(0)
Jun 29 10:48:49 osmc kernel: EXT4-fs error (device sda1): ext4_ext_check_inode:495: inode #860: comm chown: pblk 0 bad header/extent: invalid extent entries - magic f30a, entries 1, max 4(4), depth 0(0)
Jun 29 10:49:27 osmc kernel: EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
Jun 29 10:49:27 osmc kernel: EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
Jun 29 10:49:42 osmc mediacenter[551]: dpkg-query: no packages found matching rbp-mediacenter-osmc
Jun 29 10:49:52 osmc kernel: EXT4-fs (sda1): error count since last fsck: 367
Jun 29 10:49:52 osmc kernel: EXT4-fs (sda1): initial error at time 1429463500: ext4_ext_remove_space:2961: inode 6767
Jun 29 10:49:52 osmc kernel: EXT4-fs (sda1): last error at time 1435567767: ext4_find_extent:900: inode 4749
[ 21.949220] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 21.951104] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 21.951771] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 21.952451] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 21.953058] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 21.953649] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 21.954258] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 21.954863] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 21.955452] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 21.959935] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 86.985930] EXT4-fs error (device sda1): ext4_mb_generate_buddy:757: group 7, block bitmap and bg descriptor inconsistent: 22164 vs 22290 free clusters
[ 87.991036] JBD2: Spotted dirty metadata buffer (dev = sda1, blocknr = 0). There's a risk of filesystem corruption in case of system crash.
[ 87.996219] JBD2: Spotted dirty metadata buffer (dev = sda1, blocknr = 0). There's a risk of filesystem corruption in case of system crash.
[ 196.628078] Console: switching to colour frame buffer device 240x67
[ 197.139612] EXT4-fs error (device sda1): ext4_iget:4056: inode #859: comm chown: bad extended attribute block 1073741824
[ 197.142658] EXT4-fs error (device sda1): ext4_ext_check_inode:495: inode #857: comm chown: pblk 0 bad header/extent: invalid eh_entries - magic f30a, entries 65, max 4(4), depth 0(0)
[ 197.144485] EXT4-fs error (device sda1): ext4_ext_check_inode:495: inode #860: comm chown: pblk 0 bad header/extent: invalid extent entries - magic f30a, entries 1, max 4(4), depth 0(0)
[ 240.265071] EXT4-fs error (device sda1): ext4_ext_check_inode:495: inode #857: comm dpkg: pblk 0 bad header/extent: invalid eh_entries - magic f30a, entries 65, max 4(4), depth 0(0)
[ 240.872810] EXT4-fs error (device sda1): ext4_iget:4056: inode #859: comm chown: bad extended attribute block 1073741824
[ 240.876325] EXT4-fs error (device sda1): ext4_ext_check_inode:495: inode #857: comm chown: pblk 0 bad header/extent: invalid eh_entries - magic f30a, entries 65, max 4(4), depth 0(0)
[ 240.878659] EXT4-fs error (device sda1): ext4_ext_check_inode:495: inode #860: comm chown: pblk 0 bad header/extent: invalid extent entries - magic f30a, entries 1, max 4(4), depth 0(0)
[ 278.035588] Console: switching to colour dummy device 80x30
[ 278.949029] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 278.955192] EXT4-fs error (device sda1): ext4_find_extent:900: inode #4749: comm kodi.bin: pblk 97509 bad header/extent: invalid magic - magic 5625, entries 63474, max 3115(0), depth 64902(0)
[ 303.902089] EXT4-fs (sda1): error count since last fsck: 367
[ 303.902125] EXT4-fs (sda1): initial error at time 1429463500: ext4_ext_remove_space:2961: inode 6767
[ 303.902145] EXT4-fs (sda1): last error at time 1435567767: ext4_find_extent:900: inode 4749
It’s difficult to tell if you just have file system corruption or whether the hardware of the drive is failing. You could try running a file system check by booting while holding down SHIFT but I suspect that it may be too corrupted for that to repair everything and/or the drive itself is faulty. Give it a try though.
Is it a thumb drive or a spinning drive ? If the latter how is it powered ? Have you had any previous unexplained crashes ? Do you always shut down properly or does it get the power pulled ?
Updating won’t have caused these errors however if there was a pre-existing hardware fault with the drive or the file system was already badly corrupted this would cause the update process to fail as lots of critical system files get written to disk during an update - and if they can’t be written correctly bad things will happen…