iMac G3 won't turn on

It is better to take your computer to the Apple Store. That smoke is not a very good sign. It can be a short circuit in the audio card, but it is impossible to tell for sure without investigation.

You are welcome, good luck!

Doubtful that an Apple Store will even look at it... Not worth the effort of hauling it there. It is considered "obsolete" (by many years in this case).

It's an old iMac with a CRT display. A CRT, unlike an LCD, requires very high voltage. The component that supplies that dangerous high voltage goes bad over time (even if you don't use it). The system is designed to shut off and not turn back on, if a power-related failure is detected, for safety reasons.

It's possible you may find a local independent repair shop with expertise in fixing CRT displays. It would be a shop that knows how to fix old-school TV sets (from before they became flat). But most likely, it's a lost cause not worth the expensive of repair.

Finding that exact CRT part in stock somewhere is probably difficult. And the problem is usually not the CRT itself. It's on the board that provides the high voltage needed to run the CRT. The repair is usually to replace (unsolder and re-solder) parts on that board (like capacitors). It's doable by a person with the technical expertise. But unless you know what you're doing, poking around in the area of the iMac that holds the CRT is dangerous. That high-voltage continues to exist, even when the iMac is unplugged from power.

The 350 Mhz iMac G3 has a slot-loading optical drive, correct? If it was the older "tray-loader" iMac G3, you can (safely) disconnect the internal CRT video cable, attach an adapter to convert to VGA, and plug it to a regular VGA display. And it may actually work (depending on exact damaged component). But for your model, it won't even power ON unless you fix the current problem.