How do I prevent doors on a media cabinet from sagging?
Ply is good in shear, helping the door from racking.

I would use euro style hinges, meant for frameless cabinets. They will transfer the door weight better than simple butt hinges.  They also have 3 way adjustments for compensating for sagging over time.  Just be sure to start out with a good gap and room for the hinge to adjust up.