When you ask for help, the trick is to make people want to talk to you. You'll get much more - and better, and more timely - help when you set up the situation so people want to spend time with you. There are certain concrete things you can do to "grease the wheels" so to speak, and make sure that you're creating a relationship where help is freely offered.
Respect their schedule.
Remember, you're the one who needs something. The other person (or group) involved is granting you something, so you need to work with them. Come in early, leave late, etc. The goal is to inconvenience them as little as possible.
Don't tell them how to do their jobs.
A really fast way to make someone defensive is to ask a favor and then proceed to lecture them about how exactly they should go about doing it. Just don't do this. Don't tell a system architect how to design a solution that meets your perceived needs - you're leaving her guessing what on earth you're trying to accomplish. Don't tell a project manager which person to allocate to fulfilling your request - you're not letting them pick the best (and most available) resource. Your goal is to describe the change you wish to see made, not how that change should be made. Telling them how to fulfill your request limits their options and you will likely not get an ideal solution.
If I had to pick one thing, this would be the top thing that really can make people avoid you instead of want to help you.
Acknowledge that it may be your problem.
Noting up front that some request may be a result of your lack of understanding, or a mistake you may have made humbles you a bit. It also prevents your request from sounding accusatory. By acknowledging that you may have caused the issue, you change the tone of the conversation from "you screwed up" to "hey, here's this wacky thing. Help!" Again, it reduces defensiveness. Plus, when it really is your problem (as happens to all of us sometimes), there's a graceful way out for everyone.
Don't ask the same question twice.
Asking for help once is perfectly wonderful. Asking for the same thing a second time generally means you didn't listen the first time. And why on earth would someone want to help a guy who didn't listen the first time around? Note that this doesn't cause a problem if the repetitiveness of the question isn't apparent to either of you when you start.
Have something to offer.
Relationships are about reciprocity. Since you're asking for help, you need to offer something. This doesn't have to be tit-for-tat at all. Just be sure that you're willing to help when someone needs you. And it never hurts to bring in brownies, or mention to your helper's boss that you really appreciated the hand!
There are two halves to any interaction between two people. Your task is to understand your role and to make sure that your half of the interaction is done effectively and with the least possible negative impact on the person in the other side. Build your reputation, don't be the cause of your own problems, be precise, and temper your speech, and you will find help