Others have mentioned this subject on the forums, such as this post:

https://groups.google.com/a/chromium.org/forum/#!topic/chromium-os-discuss/KCXncqE2SDc

which discusses that currently a custom kernel is required:

https://github.com/docker/docker/issues/15679

https://gist.github.com/christianbundy/ba62890a7c2f8128bcbb

The diff in this second link indicates just a few kernel features to be enabled in the kernel config before compile.  The following are listed as changed, though I think that the ones that move from =m to =y are not necessary.

-# CONFIG_BLK_CGROUP is not set
+CONFIG_BLK_CGROUP=y

-CONFIG_BRIDGE=m
+CONFIG_BRIDGE=y

-# CONFIG_CGROUP_DEBUG is not set
+CONFIG_CGROUP_DEBUG=y

-# CONFIG_CGROUP_PERF is not set
+CONFIG_CGROUP_PERF=y

-# CONFIG_MACVLAN is not set
+CONFIG_MACVLAN=y

+CONFIG_MEMCG=y
+CONFIG_MEMCG_SWAP=y

-CONFIG_NF_NAT=m
+CONFIG_NF_NAT=y

-CONFIG_NF_NAT_IPV4=m
+CONFIG_NF_NAT_IPV4=y

(again, don't think s/=m/=y/ is needed)

A recent development is that Google will have Play Store, i.e. Android app, compatibility in future ChromeOS builds.  Does that mean that some of the needed kernel features will be enabled for the Android support?  Some discussion has hinted at that.

https://productforums.google.com/forum/#!topic/chromebook-central/Pd0CY3oi1cA

Activating. Please assign to the right owner and the appropriate priority.

This issue has been Available for over a year. If it's no longer important or seems unlikely to be fixed, please consider closing it out. If it is important, please re-triage the issue.

Sorry for the inconvenience if the bug really should have been left as Available.

For more details visit https://www.chromium.org/issue-tracking/autotriage - Your friendly Sheriffbot

This feature request is still applicable - can it be kept open? With all the recent news of vm (crosvm, crostini) and container (oci, lxc) support in development, and with various parties now able to get containers running in developer mode or in dev channel releases, I am hopeful that practical options for running Docker will be available soon.