Can I access fargate's container?

You cannot access the host in Fargate and run docker exec from there. However, you can enable ECS Command Execution in your task/service definition and then use the command aws ecs execute command which is going to have the same effect.

I can't recommend more to configure your cluster logging settings to log these exec commands so that you can audit when your users ran commands and see what these are (and their outputs).

Activating it is rather simple, but if you want something that will take care of creating the task definitions and possibly the cluster for you, look at x-cluster and x-ecs.EnableExecuteCommand which will create the CFN templates for you.

If you are using an existing cluster and these settings are set for command logging, compose-x will automatically detect that and configure IAM permissions accordingly.

Hope this helps,