For Checking-In it can be possible if you Enable Moderation and Enable Versioning in your list/library.. So once people Check-In the item, the changes will be visible to all approvers and once they review the changes (they can see Check-In comments in Version History), they can Publish the item to make the changes available to all users..
Now a problem with this is that Check-In comment is not by default REQUIRED, it can be left blank.. For that you can write an Event Receiver as explained here
About approval before a Check-Out, I don't think its OOB possible.. You might need to write your own custom Ribbon Action (Custom Check-Out button), that lets say open a Dialog for user to enter comments (this can be a Visual Web Part placed on Page opened as a Dialog).. And you can email this request to approvers, you will need to have a separate page for approvers to Reject/Approve the request.. Once they approve you can programmatically Check-Out the item on Requester's behalf..
All theoretical, let me know if you need help somewhere