You were probably foolish to try this before a holiday weekend. Repeat after me...
Q: What is the first rule of upgrading OS X Server on a Production Server?
A: You don't upgrade OS X Server on a Production Server.
Q: What is the second rule of upgrading OS X Server on a Production Server?
A: You DO NOT upgrade OS X Server on a Production Server without a backup!
You get the point. Hopefully, this made you smile a bit. I feel your pain.
So, first... Tell me you have a backup of the 10.6.8 boot volume before your started and that you can restore, walk away slowly, and pretend this never happened.
If this is not the case, the next step is to define a list of all services that you had running on the server before attempting the upgrade. Remember, there are a lot of services removed between 10.6 and 10.9 (Podcast, MySQL, Tomcat, QTSS, etc). If you were using one of those services, the upgrade process may be halting. Likewise, if you are using something like mail, there may be a long reindexing process. So, while the 1 to 1.5 hours sounds very long, it is possible that the server was doing something. That leads to the next place to look.
What was being reported in the log files. System log, Open Directory log, and even the all events log can be helpful in tracking down an error or long running process.
Likewise, what does Activity Viewer show during this upgrade. What process(es) is/are consuming CPU? Disk I/O? The assumption is that the server is trying to do something (hopefully).
You are going to need your put on your Sherlock Holmes' deerstalker for this one. I will also point out that the "out of disk space" error is sounding particularly terrifying.
You may be having a very long Friday.
R-
Apple Consultants Network
Apple Professional Services
Author "Mavericks Server – Foundation Services" :: Exclusively available in the iBooks Store