The influence of different heating processes of a fibrous cellulose precursor together with the application of anisotropic mechanical stretching has been studied for the Li storage in the obtained hard carbons. A reversible capacity up-to 470 mAh/g has been reached and its origin has been associated with a large plateau at low potential vs Li. The Li-7 NMR spectrum recorded on the fully lithiated sample at room temperature presents a peak at 66 ppm. At lower temperature down to 230 K, this contribution is split into a temperature-independent peak centere at 18 ppm and an ill-defined paramagnetic shift. On the basis of the spin-lattice relaxation times, it is shown that the temperature-independent Li sites exhibit a small deviation to the Korringa type behavior at low temperature, that is associated with a distance between Li sites of the same order as for Li-Li in the metallic state. The paramagnetic NMR contribution has been related to the lithium ions inserted below 100 mV vs Li within the hard carbon. The reversibility of the electrochemical process, and especially the polarization between insertion and removal of the Li ions, is found to be related to their chemical nature, different if they are located in the nanopores or between the graphitic type layers. The data for inserted lithium in these samples clearly shows that extended delocalization of conduction electrons is not occurring.