In order to win in the market competition, enterprises have considered sustainable high-quality development (SHQD) as an important goal. However, prior studies on SHQD mainly emphasized theoretical discussions, and few scholars have conducted quantitative data analysis, especially in the state-owned enterprises (SOEs) area. Given this research gap, this paper proposes a novel framework to evaluate and select state-owned enterprise with sustainable high-quality development capacity (SHQDC) in China under linguistic distribution assessment (LDA) environment. This study has been done in four stages. In the first stage we present a detailed evaluation index system for SHQDC. In the second stage, to determine the comprehensive weights of evaluation criteria and sub-criteria, we extend the traditional fuzzy analytic hierarchy process (FAHP) method to determine the subjective weights of criteria and sub-criteria, and propose a novel method to compute the objective weights of criteria and sub-criteria under linguistic distribution assessment environment. In the third stage, we extend the traditional bidirectional projection method to the linguistic distribution environment and further put forward a ranking method to select SOEs with SHQDC, in which the ratings of sub-criteria are assessed in linguistic distribution assessments to reflect the qualitative evaluation of experts’ subjective opinions. Finally, we demonstrate the validity of the proposed approach by means of comparing the sustainable high-quality development capacity of three state-owned enterprises in China.