Recover data from unbootable android 5.0 note 3
If I understand correctly, you can only boot your device into the fastboot mode and nothing else, right? Did you try OEM (Samsung) stock recovery by pressing UpVolume+Power button and had any success?

If not, then here is one way to recover your data, but you'll need one thing for this: You should have the CWM (ClockWorkMod) recovery image for your device, or you should be able to download it from somewhere. Assuming that you can do that, here is what you can do:


Boot into the fastboot prompt and connect it to your PC's USB port.
On the PC, ensure that ADB drivers for your device are installed correctly (If you were able to run adb/fastboot commands, that's enough).
Use fastboot command to temporarily boot your device using the CWM image:

fastboot boot /path/to/CWMRecoveryImage.img

Once inside CWM recovery, adb will start working!
Just pull the files/folders you need using the adb pull command:

adb pull /sdcard/foo/bar/myfile1.dat
adb pull /sdcard/foo/bar/myfile2.dat
adb pull /sdcard/foo/baz/*



Alternatively, you can use a tool like DroidExplorer in step 5 to recover your data more easily. Hope this helps.

edit

I found this XDA link where some CWM recovery images are available for the Galaxy Note series, but be careful that you select the one for your exact model.