Someone else mentioned Cowboy Bebop, and I looked into a plot summary. I don't think it's it, but I'm going to watch that particular episode in about 10 - 20 minutes to see if it is the one. (You're talking about episode 10, Ganymede Elegy, I think.) Thanks for the info.
Ok, I watched the episode, and it's not it. Either I'm going crazy and dreaming animes up, or the anime I'm looking for isn't that popular. I had thought that it was set in Samarai-esque times, but I might be wrong about that. If anyone else has any ideas, please let me know.