Mavericks Time Machine is stuck in the middle of backing up, other times stop unexpectedly
To summarize how I worked around the problem:


made sure the problem only exists for Mavericks: made sure that another Mac (pre-Mavericks) still backs up to QNAP NAS seamlessly 
made sure that problem is only related to QNAP NAS and Mavericks together, not to Mavericks alone. For this, performed back up to external HDD

when external HDD backup got stuck with progress bar frozen, Standby-Resume of Mac helped to "unfreeze" Time Machine most of the time

to achieve first successful backup with NAS, I first excluded every folder that is on "Macintosh HD": Time Machine Preferences > Options > Exclude these from backups > +. Having minimal successful backup, I added folder by folder, from smallest ones (all &lt;1MB, then all &lt;1GB, then one by one more and more large, largest in the end.

when backup stopped itself long before the estimated time, it turned out that it actually copied the part last displayed on progress bar. The next time you invoke Back Up Now, the total amount of files to back up becomes smaller by the amount reached previous time.

for QNAP NAS, installed QFix (reboot OS X before invoking new backups). QFix seriously improved copying speed and made the speed consistent, plus resolved "frozen progress" problem at least for a full backup from the scratch.


Still, the problems left for Apple and QNAP to solve are:


progress bar which frequently gets frozen in the middle of copying
stuck "Preparing backup..." after QFix before OS X reboot
backing up stopping itself in the middle of copying, and/or "I/O error" which forces backup progress to stop