What model eMac is this?
Depending on model, eMacs can be upgraded to OS X 10.4.11 Tiger or OS X 10.5.8 Leopard.
This is a far as you can go with updating the operating system.
To update to these versions, you will need to find, purchase and install Apple retail install CD/DVDs.
These are really old OSes and are very hard and expensive to find.
Here's a sit that's not too bad for pricing on Tiger or Leopard.
Also, depending on eMac model you have, it would be wise to completely Max. out the RAM.
Early eMacs can take up to 1 GB of Ram. Late eMacs can take up to 2 GBs of RAM.
Depending on your eMac model, you may be able to find, purchase and install either an Airport or Airport Extreme wireless card into your eMac, also.
This is a much updating as you can do to eMacs.
You won't be able to update any installed software as you need OS X version greater than what you are running.
And you need to find these versions without using software update.
Software Update may tell you what version you need, but the link to that version on Apple's site may no longer exist.
You need to go into the support sections of each piece of software on Apple's site and do a search for the version of the software you are looking to update.
Well, you can't updates any of those for 10.3.9 at all, you need a higher OSX, but even then you'll still be years behind.
I think the most modern Browser for Panther/10.3.9 is Camino...
Have read that version 1.6.11 works also.
Or iCab 4.8...
No idea if this works in Panther, but...
The last really supported Flash for PPC was 10.1.102.64, but if it's for like Facebook or such, people have been fooling FB to think they have a later version installed.
Texas Mac Man's Flash hack/post...
Flash player 11.1 hack on PowerPC - https://discussions.apple.com/message/16990862
See in each Browser which version of Flash it thinks it has...
To use Mac OS X 10.4 Tiger, your Macintosh needs:
* A PowerPC G3, G4, or G5 processor
* Built-in FireWire
* At least 256 MB of RAM (I recommend 1GB minimum)
* DVD drive (DVD-ROM), Combo (CD-RW/DVD-ROM) or SuperDrive (DVD-R) for installation
* At least 3 GB of free disk space; 4 GB if you install the XCode 2 Developer Tools (I recommend 20GB minimum)
See Tom's, (Texas Mac Man), great info on where/how to find/get Tiger...
Or Ali Brown's great info on where/how to find/get Tiger...
* Mac computer with an Intel, PowerPC G5, or PowerPC G4 (867MHz or faster) processor
minimum system requirements
* 512MB of memory (I say 1.5GB for PPC at least, 2-3GB minimum for IntelMacs)
* DVD drive for installation
* 9GB of available disk space (I say 30GB at least)
Trouble is Apple no longer sells it, check eBay & such for the Retail version, not the Gray Discs...
There are workarounds if the 867MHz CPU is the only hangup, (not Bus Speed)…
List of Applications Not Compatible with Leopard...