I always build my own PCs... I also tend to upgrade my PCs one piece at a time instead of starting from scratch.
eMachines should be fine for people that don't want to mess with computers, but I wouldn't advise swapping out hardware. When it's time to upgrade, expect to upgrade the entire computer, case and all. The monitor and printer might be reusable.
Computers are getting harder and harder to build by hand, and the components are all upgrading faster than before. Now when you want to upgrade your CPU, you need to upgrade the motherboard, which requires you to upgrade the memory. Then to take advantage of the high speed IDE bus, you need to get a new hard drive. At that point, you practically have an entirely new computer anyway.
Any modern computer you can get for under $300 should be adequate. If it completely craps out on you in a few years, spend another $300 and buy a new one. You'll spend that much anyway if you try to keep a computer up to date one component at a time.
I'll never get an eMachine by choice because I do mess with my own computers, and I HATE messing with proprietary hardware. It's worth considering if you never want to open the thing or replace anything.