The conversion of any building into a dwelling, or the conversion of any dwelling so as to accommodate an increased number of dwelling units or families, shall be permitted only within a district in which a new building for similar occupancy would be permitted under the chapter, and only when the resulting occupancy will comply with the requirements governing new construction in such district with respect to minimum lot size, lot area per dwelling unit, dimensions of yards and other open spaces and such further requirements as may be specified hereinafter within the section applying to such district.