。
The refugees were living in a cramped, dingy back alley.
Literal
Refugees [subject-が] cramped-and-dingy back-alley [at-に] were-living.
Short, atmospheric. ごみごみした is a 擬態語-derived adjective — ごみごみ is a mimetic for 'cluttered, cramped, dingy' (related to ゴミ 'trash'), and + した turns it into a modifier ('crammed-together,' 'dingy and cluttered'). Japanese has a whole family of mimetic-derived adjectives formed this way: のんびりした (laid-back), しっかりした (solid, reliable), はっきりした (clear). The verb 住んでいました uses ~ている in its resultative-state sense for a verb of residence — 'was living, as an ongoing state.' 裏町 ('back town,' 'back alley') is a compound for an unglamorous, often poor neighborhood.