Put iMac G3 in Target Disk mode
Only FireWire and Thunderbolt support Target Disk Mode.
You have to remove the internal disk and put it in an external case with UltraATA interface.
Alternatively you may create a bootable thumb drive (MacOS 9.1 or better), boot from it and create an image of your HDD with Disk Copy and save it to a network share. Building a bootable thumb drive and booting from it may take a while though because of the low bandwidth of USB1.1 (12 MBit/sec ≈ max. 1MByte/s).
According to this article Mac OS X 10.3.9 - the latest Mac OS X which supports the G3 - unfortunately doesn't provide USB-booting:

Further refinement I just came across in my notes, on the system requirements to boot from Macintosh USB ports: OS 8.6 through 9.2.2, and OS 10.4.3 and later. OS 8.6 - 9.2.2 might need the last of whichever versions of the USB extensions that each of these System versions supports (for OS 9.1 through 9.2.2, USB 1.5.6 is the preferred version; OS 9.1 and later don't need the extension &quot;USB Mass Storage Support&quot;, especially since it sometimes freezes Macs running OS 9.1 - 9.2.2 that are trying to boot from USB), though other details as to the proper combination and versions of USB extensions under OS 8.6 - 9.2.2 is screwy, and too lengthy to post here, especially since we're dealing with OS X anyway.
The first Macs to support USB booting, are the first slot-loading iMacs (including the 350 MHz model with no Firewire ports), and the Firewire Powerbook G3. I've even booted OS 10.4.11, from a USB 2.0 flash drive, on a Mirrored Drive Doors G4, from a USB 2.0 PCI slot card, with no special tricks, Open Firmware modifications, etc.