The hierarchical packaging of chromatin fibers plays a critical role in gene regulation. The 30-nm chromatin fibers, a central-level structure bridging nucleosomal arrays to higher-order organizations, function as the first level of transcriptional dormant chromatin. The dynamics of 30-nm chromatin fiber play a crucial role in biological processes related to DNA. Here, we report a 3.6-angstrom resolution cryogenic electron microscopy structure of H5-bound dodecanucleosome, i.e., the chromatin fiber reconstituted in the presence of linker histone H5, which shows a two-start left-handed double helical structure twisted by tetranucleosomal units. An atomic structural model of the H5-bound chromatin fiber, including an intact chromatosome, is built, which provides structural details of the full-length linker histone H5, including its N-terminal domain and an HMG-motif-like C-terminal domain. The chromatosome structure shows that H5 binds the nucleosome off-dyad through a three-contact mode in the chromatin fiber. More importantly, the H5-chromatin structure provides a fine molecular basis for the intra-tetranucleosomal and inter-tetranucleosomal interactions. In addition, we systematically validated the physiological functions and structural characteristics of the tetranucleosomal unit through a series of genetic and genomic studies in Saccharomyces cerevisiae and in vitro biophysical experiments. Furthermore, our structure reveals that multiple structural asymmetries of histone tails confer a polarity to the chromatin fiber. These findings provide structural and mechanistic insights into how a nucleosomal array folds into a higher-order chromatin fiber with a polarity in vitro and in vivo.