You don't need to add 5gb space to Linux.
What you need to do is finding the large file.
If those are not important remove them and if those are important move them to USB. It will free up space and you can boot into your os.
To do this first you need to nevigate to the mount point and then run the command to see which files are large and taking most space.
Code:
du -ah . | sort -rh | head -n 20
After the temporary solution you need an external hard drive to move your data.
It will help to solve your two problem
1. It will free up enough space to run os smoothly
2. Your data will be backed up in a safe storage.