With the advance of the thermoplastic plastic elastomer (TPE) technology, there are growing interest and needs for using these materials in the meltblowing process where benefits of small fiber diameters of meltblowns can be combined with rubber‐like elastic properties of elastomers. Performances and utilities of wide ranges of meltblown products such as facemask, medical barrier, wound‐care, diaper can be drastically improved with additions of TPE. In this study, a new elastomeric meltblown fabric was successfully made with the styrene–ethylene/butylene–styrene (SEBS) block copolymer, and the relationship among structure, tensile properties, and meltblowing process parameters are studied. We found that median fiber diameter increases with the polymer mass throughout and decreases with air pressure, and fabric solidity has significantly influenced by die collector distance (DCD). The pore sizes of the fabrics are directly influenced by fiber diameters at the given DCD, but higher DCD increases the pore size due to their open structures. All SEBS nonwovens exhibit high strain at break, larger than 400%. Processing parameters significantly affect tensile properties, and this can be attributed to the fabric structure changes. The reduction of fiber diameter tends to increase the tensile strength of the fabric as it created more fiber‐to‐fiber bond points.