To see if Docker has started, do this command:
ps -ef | grep -i docker
If that returns only a service for the grep itself, then Docker is not running. Occasionally the Docker service won't start through traditional methods. But some users have found that this command will work reliably:
docker daemon &
The "&" allows for the next prompt to return. This method is explicit to new users of Docker too. This method provides more verbose informational messages to print to the console as compared to "systemctl docker start."