The Greek word ‘sarx’ (flesh) is often a difficult word to translate and define in Paul’s epistles. Most Bible translations use more than a few English words for the Greek word ‘sarx.’ For example, some translations use “human body,” “body,” “person,” “sinful flesh,” “earthly,” “physical,” “natural,” or other similar words to translate ‘sarx.’ So what does this word mean? It is a long answer, I suppose, since the term has various meanings depending on context. I appreciate Douglas Moo’s summary of this term in his article, “Sin in Paul.” Here’s how Moo summarizes the meanings of ‘sarx.’
1) The most basic meaning of sarx, and the most common in secular Greek, is ‘the material that covers the bones of an animal or human body.’ Paul occasionally uses the word with this sense (cf. 1 Cor. 15:39, Eph, 2:11, Col. 2:13, Gal. 6:13).
2) Following precedents in secular Greek, Paul also applies sarx to the human body as a whole (cf. 2 Cor. 7:1, Gal. 4:13, Eph. 5:31).
3) But more often, Paul uses sarx to refer not to the human body narrowly but the human being generally (cf. 1 Cor. 1:28-29, Gal. 1:16, 2:16).
4) This #3 meaning merges almost imperceptibly into a bit broader concept, namely, the human state or condition. While debated, 1 Cor. 10:18 probably falls into this category. This is what some call the ‘neutral’ use of sarx. [Although some scholars say] a certain negative nuance often clings to sarx even when Paul uses it in apparently neutral senses (cf. Rom. 1:3-4, 9:5).
5) Finally, sarx can designate the human condition in its fallenness (Gal. 5:16-17). This is what some call the ‘ethical’ use of sarx. This sense of sarx is quite common in Paul.
[As a side, this reminds me of Ridderbos’ explanation of ‘sarx’: “On the one hand, ‘flesh’ has for [Paul] the significance of what is human in its weakness, dependence on God, and perishableness in itself; on the other hand, ‘flesh’ is the pregnant and very specific description of man in his sin, and the coinciding of being human and being a sinner is therefore expressed in it” (Paul: An Outline of His Theology, p. 93). It seems like Ridderbos is also working with a “neutral” and “ethical” sense of the word ‘sarx.’]
While there is more to this discussion, and while this may not answer all the questions about the term ‘sarx,’ it is a helpful outline to consider when thinking about this word in Paul’s epistles.
The above outline (which I’ve edited slightly) is found in Moo’s article in Fallen: A Theology of Sin.