Topological semimetals (TSMs) in which conduction and valence bands cross at zero-dimensional (0D) Dirac nodal points (DNPs) or 1D Dirac nodal lines (DNLs), in 3D momentum space, have recently drawn much attention due to their exotic electronic properties. Here we generalize the TSM state further to a higher-dimensional Dirac nodal sphere (DNS) or pseudo DNS (PDNS) state, with the band crossings forming a 2D closed or approximate sphere at the Fermi level. This new TSM state can exhibit unique electronic properties, making DNS/PDNS a new type of fermion beyond DNP/DNL paradigm. In the realistic crystals, we demonstrate two possible types of PDNS states underlied by different crystalline symmetries, which are characterized with a spherical backbone consisting of multiple DNLs and approximate band degeneracy in between the DNLs. We identify all the possible band crossings with pairs of 1D irreducible representations to form the PDNS states in 32 point groups. Importantly, we discover that strained M H3 (M = Y, Ho, Tb, Nd) and Si3N2 are materials candidates to realize these two types of PDNS states, respectively. As a high-symmetryrequired state, the PDNS semimetal can be regarded as the "parent phase" for other topological gapped and gapless states.