But if the wire is not connected to the terminals of the emf, they will pile up at one end of the wire, leaving net positive charge at the other end. The separated charges in the wire then produce their own electric field opposing the field produced by the emf. The interior of the wire will be field free and charges will no longer accelerate. If, however, the wire is connected to the emf, then the emf can pump the electrons through its interior back from the positive to the negative terminal. Then field inside the wire will not be zero, and electrons will continue to accelerate towards the positive terminal.
Steady currents can only flow in continuous loops.
At any point, just as much charge has to flow out of a small volume surrounding the point as flows into the volume. If this were not so, charge would accumulate at the point, setting up its own electric field. This field would exert an additional force on the moving charges, disrupting the steady current. The electric field in a homogeneous wire with constant cross-sectional area carrying a steady current is the same everywhere. If it were not, electrons would move with different velocities in different sections, and charges would accumulate in certain regions.