Depending upon the specific model you have you can upgrade to either 10.4.x or 10.5.x. Neither are available from Apple any longer, so you will need to look to eBay or Amazon for copies. Be sure to only purchase retail copies and not gray labled machine-specific copies.
If your eMac came with 10.3 originally and has a clock speed of 1 GHz or higher, then you can install Leopard, 10.5.x.
If you don't want to lose data then make backups.