I don't have a car or anything — I'm getting by on welfare.

Literal

Car [also-も] anything not-exist, welfare [by-で] am-living.

A stark personal-circumstances statement. ~も~もない ('neither X nor Y,' 'don't have X or anything') is a coordinated negative that emphasizes absence. 車も何もなく combines this with the written connective form of ない (→ なく, the stem-linking form). 生活保護 ('livelihood protection') is the Japanese term for welfare / public assistance. 生きる ('to live') with ~ている ('am living') marks the ongoing state of getting by. The で marks the means — 'by means of welfare.' The whole sentence conveys a deeply modest circumstance in bare terms.