To process data during the backup, e.g. apply compression, encryption, or perform byte-level comparison for differential backups, Handy Backup actively uses temporary files. For operations involving large sets of data, there may be not enough space to create a temp file, which will result in the above error.
If you see this error, you can change the location of your temporary directory (this is done in Settings -> General Settings) or divide the task into several smaller ones.
See also: FAQ | Backup Errors: