Galaxy S3 GT-I9300 Stuck at Samsung logo with no recovery and no download mode
Does your volume up button work? If i'm not wrong, Download mode never fails to boot into. 
Let's assume it is not working and in this case, you'll need a way to force boot in download mode.
There are two methods I could find:


Using a USB JIG (and here)
Using a software called Samsung 300K Tool (Video)


After that, flash stock firmware and everything should work now.

Other solution would be replacing the volume buttons flex cable or making jumpers from the volume contact to the motherboard. Schematic below:



And 


Source: gsmhosting.com