The top answer works in the case that you want to examine the state immediately prior to the failed command.
However, the question asks how to examine the state of the failed container itself. In my situation, the failed command is a build that takes several hours, so rewinding prior to the failed command and running it again takes a long time and is not very helpful.
The solution here is to find the container that failed:
$ docker ps -a
CONTAINER ID IMAGE COMMAND CREATED STATUS PORTS NAMES
6934ada98de6 42e0228751b3 "/bin/sh -c './utils/" 24 minutes ago Exited (1) About a minute ago sleepy_bell
Commit it to an image:
$ docker commit 6934ada98de6
sha256:7015687976a478e0e94b60fa496d319cdf4ec847bcd612aecf869a72336e6b83
And then run the image [if necessary, running bash]:
$ docker run -it 7015687976a4 [bash -il]
Now you are actually looking at the state of the build at the time that it failed, instead of at the time before running the command that caused the failure.