We propose that the coexistence of gapless Fermi arcs with the metal-insulator crossover of in-plane resistivity in strongly underdoped cuprates can be explained by the FL* fractionalized Fermi liquid nature of these systems and it is not due simply to disorder effects. The particle excitations of this FL* system derived from variants of the t-J model are: the gapless holon carrying charge with small Fermi momentum proportional to the doping, the gapful spinon carrying spin 1/2, gauge fluctuations coupling them and the hole as a spinon-holon bound state or resonance due to gauge binding with a Fermi surface respecting the topological Luttinger theorem. In our proposal Fermi arcs are determined by the hole resonance, whereas the metal-insulator crossover is dominated by spinon-spinon (with subleading holon-holon) gauge interactions and this dichotomy is able to explain their coexistence.