CRT monitor not starting up but computer is!

This same thing happened to my eMac when it was 6 months old. Fortunately it was under warranty and Apple replaced about $500 worth of components. It's been fine ever since (18 months).
Check out some other threads here about the eMac video problems Apple is repairing.
Good luck.

eMac 1.25 gHz Mac OS X (10.4.4) 1.5 MB RAM, 40 GB internal drive, 160 GB firewire external drive

It's worth checking to see if your eMac is covered under eMac Repair Extension Program for Video and Power Issues. Otherwise, you can use the miniVGA adapter to plug any standard VGA monitor into the eMac's miniVGA port to mirror the vanished internal display. If you ask around friends and local small offices, odds are high that several of them have old VGA CRTs stuck in the back of a closet waiting for someone to get around to hauling the thing to a recycling center, so you could easily oick up a VGS CRT monitor for the asking.