How to backup files from Disk Utility in Recovery Mode.

Technically it is possible to recover data while booted into Recovery Mode if you know how to use the command line (assuming the data volume can be mounted). If you are not familiar with the command line, then you have two other possibilities. You can try installing macOS to a spare external USB3 drive, or you can put the Mac into Target Disk Mode so the Mac appears as an external drive when connected to another Mac (again assuming the Data volume can be mounted).

Keep in mind that First Aid only checks the integrity of the file system and partitions. First Aid does not check the physical health of the drive. In fact, First Aid does not always tell the truth about the integrity of the file system so even if First Aid says everything is "Ok" you should click "Show Details" and scroll back through the report to see if any unfixed errors are listed. If there are any unfixed errors listed with an APFS file system, then you will need to erase the drive and restore from a backup or clone. It is very unfortunate that First Aid reports the file system status incorrectly.

You should run the Apple Diagnostics to see if any hardware issues are detected with the Mac. Unfortunately the diagnostics don't detect most drive failures although there are other ways of checking the health of a drive especially if you have a full macOS bootable external drive, or a bootable Linux USB stick can be used with the correct instructions & guide.

Since the "MacHD - Data" volume is grayed out, then that means the volume is not mounted so you will be unable to access the data on it since that is where your home user folder(s) are located. You will either need to unlock (If encrypted), mount, or repair the volume in order to mount it. You can attempt to mount it (even unlock it) while booted into Recovery Mode by using Disk Utility. Otherwise, you would need to use a data recovery app such as Data Rescue while booted from an external macOS USB drive or while the Mac is connected to another Mac in Target Disk Mode.

Keep in mind that erasing the drive will destroy all data on the drive so make sure you have exhausted all methods of recovering the data before erasing the drive. Also, it is very important to determine the health of the drive. Attempting to repair or recover data from a failing drive will end up making the drive failure worse to where even an expensive professional data recovery service will be unable to recover any data. Most normal apps (even data recovery apps) should not be used on a failing drive because those apps are unable to handle all the errors produced by a failing drive which makes the drive failures worse. macOS also has lots of problems as well dealing with failing drives.

If this is all too much or confusing regarding recovering your data, then you should take your Mac to an Apple Authorized Service Provider who may be able to assist you (hopefully they have a tech who understands to check for the health of the drive before blindly trying to fix the file system or using a data recovery app). An Apple Store is not the best option if a drive is failing. I see too many techs overlook a basic physical drive health check before proceeding with data recovery which can be a fatal mistake to the data. If a drive is confirmed to be physically bad, then you should contact a professional data recovery service to attempt the recovery of your data unless you deem the techs are data recovery experts with failing drives.

Of course if you have given up on recovering the data, then you can use Internet Recovery Mode to erase the drive (or at least the Container) and reinstall macOS which will destroy all data on the drive. In theory just erasing the "MacHD" volume and selecting that as the destination should not affect the actual user data stored on the other "MacHD - Data" volume, but this is still a lot of risk to the data doing it this way.

You should always have frequent and regular backups of your computer and all external media (including the cloud) which contains important & unique data. Waiting until your computer has a problem is the wrong time to think of backups and is many times fatal to the cherished data.

Don't apologize for asking questions. That is how we know that you do not understand something. I guess I did not make something clear enough. Know of no way that you can recover your files in the Recovery Mode. It is only designed to help you do repairs when they are needed. What I said above is still what you must do. Here is the sequence I suggest.

• Open Disk Utility and attempt to have it repair your hard drive. If it does, then you will be able to restart the computer normally.
• Next, restart the computer in the Internet Recovery Mode. (Hold down the Command + Option + R keys at startup.) This will enable you do download and install the computer's original operating system. Using that, you wshould then be able to restart the computer normally.
• Once you have restarted the computer, you may then proceed with opening the Time Machine app and then transferring your files to the backup drive.

I know of no other way to allow you to transfer data to the backup disk. you seem to want to do it in the Recovery Mode and that will not work. Recovery, in this instance does not mean the recovery of your files. It is designed to either repair the computer's operating system or to reinstall a new, clean operating system so the computer can be started up. Follow the steps I have recommended. I hope it all works for you.

So far as I know, if the Mac cannot be started up there is no way to copy any files which are on its disk. Your only option is to take the computer to a recovery service which specializes in doing this and having it done professionally, Be prepared, this is a very expensive process. Now you know why having a computer backed up is so important.

Being able to boot up in the Recovery Mode is a good thing. What it will enable you to do is to first open Disk Utility and attempt to have it repair your hard drive. Sometimes Disk Utility must be run several times to complete the repairs - if they can be done. If not, then you should have the ability to restart the computer in the Internet Recovery Mode. (Hold down the Command + Option + R keys at startup.) This will enable you do download and install the computer's original operating system. If that is successful, then you will be able to run Apple's Time Machine to backup to the external disk. You still have hope. Good luck.