Why does a Time Machine backup not complete when destination is a Shared Volume?
This question requires additional troubleshooting to pinpoint the problem. But there is probably a bottleneck somewhere. 

There are a few things you don't tell us in your question. How are the different computers connected?


WiFi or Ethernet?
What speed is the link to the eMac 11/54/100/360/1000 Mbps
what are the network devices used to interconnect the devices (hubs, switches, routers...)


Make sure all devices have their own sparse bundles on the hard drives.

You should then try to test the different features used during the time-machine backups.


Test the speed of your hard drives and then during backup monitor the disk usage
monitor the memory and processor usage on your eMac
monitor the connection usage (bandwidth)


During the monitoring process try to add more and more macs until you get the errors.

If you see any of these components saturating during the backup you will pinpoint the bottleneck(s).