We develop a formalism for constructing particle-number-conserving Gaussian fermionic projected entangled pair states [U(1)-GfPEPSs] and show that these states can describe ground states of band insulators and gapless fermions with band touching points. When using them as variational Ansätze for two Dirac fermion systems (the π-flux model on the square lattice and the [0,π]-flux model on the kagome lattice), we find that the U(1)-GfPEPSs, even with a relatively small bond dimension, can accurately approximate the Dirac Fermi sea ground states. By applying Gutzwiller projectors on top of these U(1)-GfPEPSs, we obtain a PEPS representation of U(1)-Dirac spin liquid states for spin-1/2 systems. With state-of-the-art tensor network numerics, the critical exponent in the spin-spin correlation function of the Gutzwiller-projected π-flux state is estimated to be η≈1.7.