I'd have to agree with the above post that recommends that you do a "manual migration" -- that is, "hand copy" the things you want to move over.
Since the g5 is a PowerMac, the internal hard drive should be easily removable, and it's SATA, right?
If this is the case, you might consider physically removing the drive from the g5, and putting it into a USB3/SATA docking station to get at your stuff. To see what these gadgets are, go to amazon and enter "usb3 sata dock" in the search box. You'll get many hits -- they can be found for as little as $20.
Put the bare drive into the dock, then connect it to the new iMac.
It should mount on the desktop, then giving you access to the contents.
If you get any permissions errors, click ONE time on the drive icon, then at the bottom of the Get Info box, unlock the lock and check the box to "ignore ownership on this volume".
Insofar as moving items in your home folder is concerned, you face some limitations. That is:
- You CANNOT simply copy the folders such as "music", "movies", "pictures", "documents", etc., BUT
- You CAN copy the -contents- of those folders (including nested folders and their contents), in most cases.
For moving applications, your best course of action is to do an assessment of which apps on the g5 are PowerPC-based, and which are "Universal Binary". There's no point in trying to move the PPC-based apps, they won't run. For the UB apps, it might be worth doing a "clean install" of these onto the new iMac, instead of trying to "migrate" them.
To see which apps are PPC and UB, run the Apple System Profiler. Click on "applications", and in the upper right, choose to sort by type. NOTE: This only works if Spotlight is enabled.
For your accounts, again it might be best to "start fresh".
All of this is more work than a simple migration using MIgration Assistant, but again, it will probably be worth your time and trouble...