Answer
a. No.
b. Yes, this is possible.
Work Step by Step
a. Assume that we have a region of space with uniform positive charge density, and uniform electric field. Consider a cubical Gaussian surface of side s, with the uniform electric field perpendicular to a face. It must contain a net positive charge by our assumption of uniform positive charge density.
The flux through that face is $-Es^2$ and the flux through the opposite face is $Es^2$. The flux through the other 4 sides is zero because the electric field is perpendicular to the area vector. In summary, the net flux through this closed surface is zero. Gauss’s Law shows that the enclosed charge is zero, which contradicts the statement that the charge enclosed is positive.
b. A large (“infinite”) charged plate creates a uniform electric field, and the charge density $\rho=0$ everywhere in space except where the plate is.
