Sagging drawer bottom
You could use anti-friction tape.
The other thing you could consider is sheet metal.  You could get 20g sheet metal and use that to make cross bridging.  It might be rigid and thin enough to help support the bottom.
Another idea would be to split the interior drawer into two sections and use the drawer divider to pull up the bottom (either splitting the space side to side or back to front).
Outside of that you are better off rebuilding the drawer and using something reasonable as a drawer bottom.  Hard to imagine 1/4 MDF being used as a drawer bottom - did they run out of cardboard?