A comprehensive first-principles theoretical study of the electronic properties and half-metallic nature of finite rectangular graphene nanoribbons is presented. We identify the bisanthrene isomer of the ${\mathrm{C}}_{28}{\mathrm{H}}_{14}$ molecule to be the smallest graphene derivative to present a spin-polarized ground state. Even at this quantum dot level, the spins are predicted to be aligned antiferromagnetically at the two zigzag edges of the system. As a rule of thumb, we find that zigzag graphene edges that are at least three consecutive units long will present spin polarization if the width of the system is $1\mathrm{nm}$ or wider. Room temperature detectability of the magnetic ordering is predicted for ribbons with zigzag edges $1\mathrm{nm}$ and longer. For the longer systems studied, spin wave structures appear in some high spin multiplicity states. Energy gap oscillations with the length of the zigzag edge are observed. The amplitude of these oscillations is found to be smaller than that predicted for infinite ribbons. The half-metallic nature of the ribbons under an external in-plane electric field is found to be preserved even for finite and extremely short systems.