Literally: red – hand – empty – fist
Alternately: Doing something on one’s own, without receiving (and often without asking for) help from anybody. Alternately, facing an enemy without any weapons but one’s fists.
Notes: This is a repetition-type yojijukugo, in which 赤手 (in contrast to the English-language idiom) refers to a bare or empty hand, and 空拳 means that you’ve got nothing in your hand, just making a fist. One variant replaces the 赤 with another 空, while another replaces it with 徒 (to), which can also be used to mean “empty.” The latter variant seems to specifically emphasize a lack of capital when starting a business venture.
This phrase comes from the famous Journey to the West (西遊記, Japanese Saiyuuki), part II. I’m a little surprised to find that this is the first time we’ve had something on the site from that source, given its widespread influence: the closest we’ve come is, essentially, its nonfictional counterpart.