Apt Update: Problema ao desvincular o arquivo… (30: Sistema de arquivos somente leitura)

Question

Tentei as soluções fornecidas por várias outras questões no AskUbuntu, mas nenhuma delas funciona como descrito abaixo.

No Ubuntu 16.04.1 LTS, quando eu corro

sudo apt update

recebemos o seguinte erro.

Ign:1 http://security.ubuntu.com/ubuntu xenial-security InRelease
Err:3 http://security.ubuntu.com/ubuntu xenial-security Release
  Could not open file /var/lib/apt/lists/partial/security.ubuntu.com_ubuntu_dists_xenial-security_Release - open (30: Read-only file system) [IP: 2001:67c:1560:8001::11 80]
Hit:2 http://nyc2.mirrors.digitalocean.com/ubuntu xenial InRelease
Err:2 http://nyc2.mirrors.digitalocean.com/ubuntu xenial InReleasetempfiles for splitting up /var/lib/apt/lists/mirrors.digitalocean.com_ubuntu_dists_xenial_InRelease
  Could not execute 'apt-key' to verify signature (is gnupg installed?)
Ign:4 http://nyc2.mirrors.digitalocean.com/ubuntu xenial-updates InRelease
Ign:5 http://nyc2.mirrors.digitalocean.com/ubuntu xenial-backports InRelease
Err:6 http://nyc2.mirrors.digitalocean.com/ubuntu xenial-updates Release
  Could not open file /var/lib/apt/lists/partial/mirrors.digitalocean.com_ubuntu_dists_xenial-updates_Release - open (30: Read-only file system) [IP: 2604:a880:0:1::4 80]
Err:7 http://nyc2.mirrors.digitalocean.com/ubuntu xenial-backports Release
  Could not open file /var/lib/apt/lists/partial/mirrors.digitalocean.com_ubuntu_dists_xenial-backports_Release - open (30: Read-only file system) [IP: 192.241.164.26 80]
Reading package lists... Done
W: chown to _apt:root of directory /var/lib/apt/lists/partial failed - SetupAPTPartialDirectory (30: Read-only file system)
W: chmod 0700 of directory /var/lib/apt/lists/partial failed - SetupAPTPartialDirectory (30: Read-only file system)
W: Not using locking for read only lock file /var/lib/apt/lists/lock
W: Problem unlinking the file /var/lib/apt/lists/partial/.apt-acquire-privs-test.IRU0Ss - IsAccessibleBySandboxUser (30: Read-only file system)
W: Problem unlinking the file /var/lib/apt/lists/partial/.apt-acquire-privs-test.xj8X0s - IsAccessibleBySandboxUser (30: Read-only file system)
W: Problem unlinking the file /var/lib/apt/lists/partial/.apt-acquire-privs-test.3xfW8s - IsAccessibleBySandboxUser (30: Read-only file system)
W: Problem unlinking the file /var/lib/apt/lists/partial/.apt-acquire-privs-test.DxtUgt - IsAccessibleBySandboxUser (30: Read-only file system)
W: Problem unlinking the file /var/lib/apt/lists/partial/security.ubuntu.com_ubuntu_dists_xenial-security_InRelease - PrepareFiles (30: Read-only file system)
W: Problem unlinking the file /var/lib/apt/lists/partial/security.ubuntu.com_ubuntu_dists_xenial-security_Release - PrepareFiles (30: Read-only file system)
E: The repository 'http://security.ubuntu.com/ubuntu xenial-security Release' does not have a Release file.
N: Updating from such a repository can't be done securely, and is therefore disabled by default.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: Problem unlinking the file /var/lib/apt/lists/partial/mirrors.digitalocean.com_ubuntu_dists_xenial_InRelease - PrepareFiles (30: Read-only file system)
W: chown to _apt:root of file /var/lib/apt/lists/mirrors.digitalocean.com_ubuntu_dists_xenial_InRelease failed - Item::QueueURI (30: Read-only file system)
W: chmod 0600 of file /var/lib/apt/lists/mirrors.digitalocean.com_ubuntu_dists_xenial_InRelease failed - Item::QueueURI (30: Read-only file system)
W: chown to root:root of file /var/lib/apt/lists/mirrors.digitalocean.com_ubuntu_dists_xenial_InRelease failed - 400::URIFailure (30: Read-only file system)
W: chmod 0644 of file /var/lib/apt/lists/mirrors.digitalocean.com_ubuntu_dists_xenial_InRelease failed - 400::URIFailure (30: Read-only file system)
W: An error occurred during the signature verification. The repository is not updated and the previous index files will be used. GPG error: http://nyc2.mirrors.digitalocean.com/ubuntu xenial InRelease: Could not execute 'apt-key' to verify signature (is gnupg installed?)
W: Problem unlinking the file /var/lib/apt/lists/partial/mirrors.digitalocean.com_ubuntu_dists_xenial-updates_InRelease - PrepareFiles (30: Read-only file system)
W: Problem unlinking the file /var/lib/apt/lists/partial/mirrors.digitalocean.com_ubuntu_dists_xenial-backports_InRelease - PrepareFiles (30: Read-only file system)
W: Problem unlinking the file /var/lib/apt/lists/partial/mirrors.digitalocean.com_ubuntu_dists_xenial-updates_Release - PrepareFiles (30: Read-only file system)
E: The repository 'http://mirrors.digitalocean.com/ubuntu xenial-updates Release' does not have a Release file.
N: Updating from such a repository can't be done securely, and is therefore disabled by default.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: Problem unlinking the file /var/lib/apt/lists/partial/mirrors.digitalocean.com_ubuntu_dists_xenial-backports_Release - PrepareFiles (30: Read-only file system)
E: The repository 'http://mirrors.digitalocean.com/ubuntu xenial-backports Release' does not have a Release file.
N: Updating from such a repository can't be done securely, and is therefore disabled by default.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: Problem unlinking the file /var/cache/apt/pkgcache.bin - RemoveCaches (30: Read-only file system)
W: Problem unlinking the file /var/cache/apt/srcpkgcache.bin - RemoveCaches (30: Read-only file system)
W: Not using locking for read only lock file /var/lib/dpkg/lock

Tentativa 1

~# mount -o remount,rw /
~# reboot

Resultado: Mesmo erro encontrado após a reinicialização.

Tentativa 2

~# sudo fsck -Af
fsck from util-linux 2.27.1
~#reboot

Resultado: Mesmo erro encontrado após a reinicialização.

Mais informações

~# dmesg | grep "EXT4-fs error"
~# mount -t ext4
/dev/vda1 on / type ext4 (ro,relatime,data=ordered)

O que está acontecendo? Alguma idéia?