Docker requires the web server to use LVM or BTRFS, neither of which is enabled on my webserver.
My webserver runs inside a VM (on top of Xen, using shared storage directly from SAN/NAS infrastructure)
Using Docker means I will need to make substantial changes to the webserver (adding and enabling docker support and LVM) and underlying infrastructure (iSCSI luns to the webserver + modifications to the backup system) to make this work which I would prefer to avoid if at all possible.