Basically, you can
Upgrading the machine:
- Upgrade the machine to a higher configuration which has more volume, moved RAM and better processor
- This will move all the disk data to the new machine without any issues.
- Now that you have more space you can use any WordPress backup plugin to backup, download, and upload (if you are a dev, you can use wget for faster transfer)
- Take the smallest (in terms of space consumed on disk), make backup, and move
- Take the next smallest and do the same
- Repeat until you get to the biggest site and you will have enough space to back and move
Fuzzy and time-consuming way
- Backing up the database from phpmyadmin or SQL command and downloading the file
- For file, Using connecting filezilla using SSH or FTP and backing up the file
Note: Unzipped file takes quite good amount of time to be downloaded. Be patient, and ensure that you have continuous internet and power backup for smoother transfer.