Biasing with Negative value of Vgs in NE76038 GaAs MESFET
Indeed Vgs needs to be negative.

If you insist on making Vs (potential at the source) zero (connect it to ground) then you'd have to use a negative voltage to bias the gate. 
This can be cumbersome if you have no negative supply rail.

But that's not needed !

What if you'd bias the gate to 0 V (GND) and "lift" the source instead. So bias the source at 0.338 V.

That also gives Vgs = -0.338 V. That is easy as you want 10mA flowing so 0.338 V / 10 mA gives .....ohms (you do the calculation).

For the signal it is still a common source due to the capacitor in parallel with Rs



You have to apply your signal around ground level so if that is not the case (it has a DC level larger than zero) use an AC coupling capacitor to apply the signal to vG. Unfortunately the circuit does not show this.