There are a couple of good sites for information about do-it-yourself washing machine repair. Ours had a leak, which I think I traced to a tiny hole in the low-pressure drain hose that was fixable with a pipe tape that the good folks at Stadium Hardware helped me find. Let's hope the repair holds!
I used the Family Handyman's How To Repair a Leaking Washing Machine as my guide. The pinhole sized leak was in the middle of the low-pressure drain hose, where a rusty piece of bracket had wedged itself against the hose and slowly caused it to get a pinhole sized hole. The repair was made with Duck Brand Wrap-Fix Tape, a self-fusing silicone tape that says it's good to 100 PSI. And Stadium Hardware (Arborwiki link) is a marvelous store, with helpful folks and a seemingly infinite supply of parts for every hardware need.
Now let's just hope the repair holds! The next step, if it doesn't, is to replace that hose entirely. Repair Clinic has about 70 different GE washing machine hose parts available; I haven't yet identified the exact part.
Please note that the repair for the high-pressure supply hose would be very different, as noted in the comment below. The Family Handyman guide referenced above suggests getting a metallic no-burst supply hose similar to this one from Fluidmaster to replace any worn, corroded, or leaky supply hoses. Supply lines run at much higher pressure than drain lines.