Unfortunately you won't be able to play Flash videos or games in a browser. Some people have luck with TenFourFox and a hack to make it play Flash, but my experience with that only gets choppy, freezing, stuttering videos, especially on a very old PPC with not a lot of RAM like your eMac. (Obviously a G5 with a lot of RAM works better than an older G4 with limited RAM) But you can play Flash videos from Youtube in MacTubes, a standalone app that works great. I set my brother's eMac up with MacTubes and it works great.
http://www.macupdate.com/app/mac/28608/mactubes
Bookmark that because they do release updates occasionally. Sometimes Youtube changes something on their website and MacTubes has problems, but the developer normally releases an update in a day or two and things work perfectly again.
Once you download MacTubes, go into the Preferences, under the Player tab, and check Video Player. Not Flash. I've had problems with it not working if I check Quicktime, but Video Player plays Flash videos just fine.
You can even download videos from Youtube with MacTubes. (play back downloaded Flash or any type of files with VLC Player
http://www.videolan.org/vlc/index.html)
If you're going to use your eMac regularly you might consider upgrading the RAM, it's very easy to do. Your model eMac uses PC133 SDRAM and the maximum it can take is two sticks of 512mb each. Maxing out the RAM will give that eMac new life. It will boot faster, open and run apps faster, etc. If you can't afford two sticks of 512mb, buying one and swapping it for the 128mb stick you have in there will still help tremendously. (Just make sure you take out the 128mb stick, not the 256mb stick)
Here is the Everymac page for your model, which is full of every spec you can think of:
http://www.everymac.com/systems/apple/emac/specs/emac_1.0.html
And here is how to install RAM on an eMac:
http://support.apple.com/kb/HT1319