What version of the eMac? Please see this Apple article about finding your eMac serial number and using it to detemine the eMac sub-model:
then tell us the sub-model name (not your serial number). Different sub-models have different upgrade possibilities.
Also, please be specific about what is slow. We have two eMacs and even the oldest one (first year of production in 2002) is very fast for word processing and spreadsheets but terrible with browsing the web.
Given that you started this thread with the Flash player update article, there is this thread which may apply to you if you have the right version of an eMac to run the operating systems here.
eMacs with 1 Ghz or higher can run Mac OS X 10.5.8, although they will lose the ability to use Classic. Here's the thread:
Some eMacs support dual booting into Mac OS 9 and Mac OS X 10.5.8: