Both answers don't seem to really address the issue, so here's my take on this.
Windows running into a bootloop can have a few causes. One is damaged system files and another is a harddisk with drive errors.
The advise to make a backup before doing anything else is definitely a good one, but given that the cause can be a failing drive, a full 1-on-1 backup is not recommended as it may trigger a read that fails the drive completely, doing more harm than good.
So copying appdata is a good thing indeed.
What you will run into is files being in use if you attempt to copy the files from your own user, or a permission error if you copy the data from a different installation or user.
This presents a problem, the data is not copyable by conventional means. If you have the means to verify if your disk is failing (being able to run a SMART check on it) that should be your first goal. This will tell you how much you can "try" with the drive. CrystalDiskInfo should be able to provide you with this information after you transfer the disk to a working system.
Based on wheter or not the disk is fine, here are my recommendations, and why I recommend it.
The disk is fine, has no smart errors
If the disk is fine, a full disk image is the best option as it leaves the data in tact and allows you to fix the problem and attempt things. Keep in mind that this is definitely not a good idea if the disk is not fine. A disk can be failing while the SMART status is still displayed as healthy. Do not just look at the status, but also check the stats and ensure that you understand the numbers (higher numbers are usually good)
If you give up on fixing the root cause and just want to reinstall windows but have a backup of your files, copying your data is the next best thing. In order to copy your users folder, you will need to give administrator access rights to your folder. This should happen automatically if you doubleclick the folder in order to access it. It will then access all files and modify the rights to get you into the folder.
If you get an access error while setting the rights, it is likely that your disk is failing and you should abort. Copy any data that does not require you to change access permissions first, then you can attempt it again, skip all errors and consider the integrity of these files failed. See also the next section.
If your drive is failing
If your drive is failing, every second you attempt to read/write from/to it, you may damage the drive further, reducing the chance of getting information off of it.
What you want to do is prioritize the data and copy the most important things first if you haven't already done so. Anything that requires special permissions, such as the user profile should be copied last, because setting access permissions requires a recursive mutation to the disk that will access all files, and can't be controlled what is and is not accessed.
You can try to double-click the users folder to gain access, it will prompt that you need to set rights, press ok, and after 5 seconds cancel the operation.
This should at least have set rights at the root folder allowing you to enter the folder, but may prompt the same error when accessing subfolders.
Using this method, you can try to copy your documents first.
AppData is a nice folder to have, but you should narrow it down to your most important folders only. This includes your email client, but you should reconsider your browser and only do that as a last copy. Keep in mind that Google Chrome stores the passwords in the profile encrypted and they cannot be transferred by just copying the userprofile. You need access to Google Chrome, then export the passwords, and import them again later which is only possible if windows works. The only thing you can save here are the tabs you had open and your bookmarks. Keep that in mind and consider if a loss of that information is acceptable.
As for your email client, keep in mind that if you use IMAP or Exchange (not pop3) you will likely have the email stored on the server and just configuring your email client again will download everything.
If you use Outlook and you have a PST file archive, then try to backup the PST file.
What if I don't copy AppData?
Appdata can contain your settings and also sometimes contains the programs. More often than not, what is inside appdata is not considered crucial. It depends on the program, but more often than not, there's not much else than program data and you don't actually need to copy things. Some programs store their settings in the registry, and other programs store the settings in user files you open/save anyway. Program speciffic settings are often not changed, and if you change them, changing them again in the future is relatively easy.
It is not going to be a huge loss if you cannot save AppData, more an inconvenience. One exception here is Google Chrome if you store your passwords inside it. Even though this is also stored in your AppData folder, they are stored encrypted, so even a backup does not give you access to these passwords.
Final note
If the harddisk is indeed failing, you should give up on fixing the problem, and instead replace the drive and install windows fresh. Don't even consider using an image backup or clone the damaged disk to the new one. You will clone the bad sectors also and even though the drive is technically fine, SMART tools will immediately start identifying the drive as failing.
