File retrieval for Emac 10.3.9 without Time Machine.

As far as restoring the eMac, it would have come with a set of optical recovery discs, for reinstalling the operating system if necessary. You would hold down the "C" key while starting up with the appropriate disc loaded into the eMac's optical drive.

These discs were for reinstalling the operating system or applications that came with it. Erasing the eMac's disk, and writing a new copy of the operating system to it, probably would NOT enhance your odds of recovering any deleted files. All of the space formerly used to store those files would be up for grabs for overwriting.

Baird86 wrote:

[...]

How do I get the Mac into restore mode?

To add to the good guidance already provided, the eMac running OS X 10.3.9 Panther has no recovery partition to boot from. Recovery wasn't introduced to the Mac until the 10.7 Lion OS was released years later. You must use the original software restore disc(s) it shipped with.

I'm not sure how you intend to recover your files. You say that you deleted the files off the eMac itself, and that external hard drive on which you put your files got dropped and the files got lost – that is, you broke the drive.

If the eMac still had copies of your files, you could attach a drive using one of the external interfaces (USB 1 and FireWire 400, or USB 2 and FreWire 400, depending on the specific model) and try to copy them off. You would want the drive to be formatted using HFS+, since the eMac doesn't understand APFS and no current Macs have code to deal with plain HFS (no "+").

But if the only copies f the files are deleted or lost, what is there to recover?