Is it possible to solder a USB cable directly to a board?
Yes, but no.
The leads are very thin, so you need to accommodate some form of stress relief, if not they will break if you sneeze in the room next-door. This stress relief will probably need as much, if not more, real-estate on the board as a connector.
Secondly most USB leads isolation are made of PVC plastic that will melt as you solder. This will almost guarantee a short-circuit sooner or later. You can get lucky and find some that are made of a more heat resistant plastic, but most of the time this is not listed in datasheets so these are tricky to find.
Thirdly you will no longer have the continuous shield around the leads, this will almost certainly degrade the performance.