First, "운전기사, 식모살이나 하는 주제에" is not a full sentence. It is like saying "When you're ..." or "With ..." without giving the main clause. The implication is "Why do you act like that (when you're only [something bad])?".
식모 is a very disparaging word for the old style live-in maid with very low pay (the word is rarely if ever used anymore because it sounds offensive). 식모살이 means living (살이) as such a maid and 식모살이를 하다 is making a living doing that menial job.
주제 here is not related to "subject" and means a shameful state of being poor, dastardly, etc. It is a disparaging expression (with a varying degree) mostly used in phrases like NOUN-인 주제에 (when you're only NOUN, where NOUN is a despised thing) or VERB-는 주제에 to suggest the person is no more than being or doing such a lowly thing, a little like "Look at you/them, you're/they're (doing) such a despisable thing". It is used when such a person expects too much for themselves or acts too out of line with their situation.
Example: 돈도 없는 주제에 고급 레스토랑만 찾는다 = He is penniless but he's still keen on going to high class restaurants.