The patriarch of this family, and his two sons, Mahlon and Chilion, don't stick around long enough to make much of an impression on us. By the time were a few verses into the story, all the gents are dead and their ladies are left to fend for themselves (taking center stage in the process).
Though the lives and deaths of Elimelech, Mahlon, and Chilion really set this whole tale in motion, we don't get any other information about what kind of men they were and how they lived their lives. In fact, the ladies seem more upset that they've been left man-less, than that they've lost these three specific guys. Naomi is bitter about her loss, sure, but she also encourages her daughters-in-law to find new husbands and quickly as possible. No time for grieving, girls, you've got to find a man. Any man will do.
Even Naomi's quest to find a husband for Ruth centers around the idea that Ruth will be carrying on her husband's name. But, in the end, Baby Obed is listed in the genealogy as the son of Boaz, not Mahlon. Geez, Mahlon can't even catch a break after he's dead.
Here, the men are really reduced to nothing more than a plot device to push Naomi and Ruth into a domestic adventure. Even the names of the two sons indicate that they won't be sticking around long. Mahlon means "weakness" or "sickness" and Chilion mean "annihilation" or "consumption" (source, p. 241). Whoa, Elimelech and Naomi were really bad at picking baby names.
If you've been paying attention up to this point in the Bible, you've probably caught on that stories about women and from a woman's point of view tend to be few and far between. But the demise of Elimelech and his sons lets us focus on the pluck and tenacity of our heroines. Yeah, it may be a man's world, but it wouldn't be nothin' without Ruth and Naomi.