This paper is dedicated to new methods of constructing weight structures and weight-exact localizations; our arguments generalize their bounded versions considered in previous papers of the authors. We start from a class of objects P of a triangulated category C_ that satisfies a certain (countable) negativity condition (there are no C_-extensions of positive degrees between elements of P; we actually need a somewhat stronger condition of this sort) to obtain a weight structure both “halves” of which are closed either with respect to C_-coproducts of less than α objects (where α is a fixed regular cardinal) or with respect to all coproducts (provided that C_ is closed with respect to coproducts of this sort). This construction gives all “reasonable” weight structures satisfying the latter conditions. In particular, one can obtain certain weight structures on spectra (in SH) consisting of less than α cells, and on certain localizations of SH; these results are new.