If the eMac is running old software like 10.3 Panther, Software Update no longer connects to Apple's server. Thye just don't support it anymore.
You can still manually download software updates yourself if you know what you are looking for, so let us know what Mac OS your eMac has, and we could provide you a few links.
To learn which Mac OS you have click on the on the menubar and then About this Mac. Post back what is says.
You are up to date with 10.4.11 Tiger. 11 was the last update.
In System Preferences > Software Update you can check on Installed Updates to see what you have. I expect you will have them all, as the previous owner probably did them.
I am not sure of the last security update but it may be
Security Update 2007-009 (OS X v10.4.11 PPC)
Note your eMac is a PPC type processor. The other one is Intel. They both can run 10.4 Tiger so that is why the two types are listed. Yours is a PowerPC Mac.
Safari for 10.4 is no longer maintained by Apple so you need to use something else.
A good web browser is TenFourFox that is still maintained by its developers.
Better security, can handle modern websites better, & has lots od Add-ons like N-Script, which I consider essential for safety these days.
10.5.8 will have some advantages & disadvantages, mostly due to higher graphic, hard drive, & RAM demands.
So we know more about it...
At the Apple Icon at top left>About this Mac, report the version of OSX from that window, then click on More Info, then click on Hardware> and report this upto but not including the Serial#...
Model Name: eMac
Model Identifier: PowerMac6,4
Processor Name: PowerPC G4 (1.2)
Processor Speed: 1.42 GHz
Number Of CPUs: 1
L2 Cache (per CPU): 512 KB
Memory: 2 GB
Bus Speed: 167 MHz
Boot ROM Version: 4.9.2f1
Well, that should be the right on...
2Z691-6037-A Mac OS X Version 10.5.0 Leopard...
Compatible with eMac, eMac (ATI Graphics), eMac (1.0 GHz G4), eMac (USB 2.0), eMac (2005), iBook G4 (Mid 2005), iMac G5 17-inch (ALS), iMac G5 20-inch (ALS), iMac G5 (17-inch iSight), iMac G5 (20-inch iSight), iMac (Early 2006 17-inch), iMac (Early 2006 20-inch), iMac (Mid 2006 17-inch), iMac (17-inch Late 2006 CD), iMac (17-inch Late 2006), iMac (20-inch Late 2006), iMac (24-inch Late 2006), iMac (20-inch Mid 2007), iMac (24-inch Mid 2007), Mac mini, Mac mini (Late 2005), Mac mini (Early 2006), Mac mini (Late 2006), Mac mini (Mid 2007), Mac Pro, Mac Pro (8-core), MacBook (13-inch), MacBook (13-inch Late 2006), MacBook (13-inch Mid 2007), MacBook (13-inch Late 2007), MacBook Air, MacBook Pro, MacBook Pro (17-inch), MacBook Pro (15-inch Glossy), MacBook Pro (15-inch Core 2 Duo), MacBook Pro (17-inch Core 2 Duo), MacBook Pro (15-inch 2.4/2.2GHz), MacBook Pro (17-inch 2.4GHz), MacBook Pro (Early 2008), Power Mac G4 (QuickSilver), Power Mac G4 (QuickSilver 2002), Power Mac G4 (QuickSilver 2002ED), Power Mac G4 (Mirrored Drive Doors), Power Mac G4 (FW 800), Power Mac G4 (Mirrored Drive Doors 2003), Power Mac G5, Power Mac G5 (June 2004), Power Mac G5 (Late 2004), Power Mac G5 (Early 2005), Power Mac G5 (Late 2005), PowerBook G4 (1GHz/867MHz), PowerBook G4 (12-inch), PowerBook G4 (17-inch), PowerBook G4 (12-inch DVI), PowerBook G4 (15-inch FW800), PowerBook G4 (17-inch 1.33GHz), PowerBook G4 (12-inch 1.33GHz), PowerBook G4 (15-inch 1.5/1.33GHz), PowerBook G4 (17-inch 1.5GHz), PowerBook G4 (12-inch 1.5GHz), PowerBook G4 (15-inch 1.67/1.5GHz), PowerBook G4 (17-inch 1.67GHz), PowerBook G4 (15-inch Double-Layer SD), PowerBook G4 (17-inch Double-Layer SD), Xserve, Xserve (Slot Load), Xserve (Cluster Node), Xserve G5, Xserve G5 (Cluster Node), Xserve G5 (January 2005), Xserve G5 (January 2005 Cluster Node), Xserve (Late 2006)
Can you possibly copy it on another Mac in Disk Utility?
Oh wait, some eMacs might not read a Dual Layer DVD, I forget the specifics.
Have you done a PRAM reset, CMD+Option+p+r...
In fact, do 3 in a row, takes a bit of time.