Proxmox container was failing to backup and from the log files the following errors appeared over and over:

• Cannot stat: Input/output error
• Cannot stat: No such file or directory

From carrying out quite a bit of research implementing the following seems to solve the issue:

Edit /etc/vzdump.conf and change the size parameter:

# vzdump default settings

#tmpdir: DIR
#dumpdir: DIR
#storage: STORAGE_ID
#mode: snapshot|suspend|stop
#bwlimit: KBPS
#ionice: PRI
#lockwait: MINUTES
#stopwait: MINUTES
size: 4096
#maxfiles: N
#script: FILENAME
#exclude-path: PATHLIST

ISSUE: Basically vzdump is running out of memory and causing the backup of the particular container to fail. By default vzdump is set to use 500MB of memory and carrying out the above forces it to use 4096MB. Obviously take a look at memory usage and make sure 4096mb is available.