Many outdoor areas that have bears have bear bag-lines setup for hikers.
In situations where the bear-bag line is not there or it is full, what dimensions should I keep in mind to protect my goodies from bears?
For instance, do I need to keep the bag at least 8 feet off the ground and 6 feet from any climbable structure?
Bears tend to jump and swipe to get your bear bag. You want above the reach of the tallest bear in the area, I'd recommend 8-10 feet at a minimum.
You also want the bag at least a bear leg's length from the trunk of a tree, probably 4-6 feet at a minimum (if possible).
Some bears will also kamikaze from the bear cables. I know of some folks who swear they've seen this, and are otherwise trustworthy, but the bear climbs to about 20-30 feet up the tree, jumps off of the tree towards the bear bags and swipes the bags on the way down. If you have one of these crazy bears near your campsite, pop some popcorn, and plan to go hungry the next day.
It depends on the area. Some places (Yosemite, Glacier Bay) there are pretty much no safe bear bags and you need a bear-proof container. The bear bag should be high enough that a bear can't reach it from the ground, and far enough from the tree trunk that it cannot reach it from the trunk of the tree.
A couple of things that are as important as the number of feet from the ground or tree:
Put everything that has an odor in a sealed plastic bag. It's best if the bear doesn't smell food from the bear bag, and this will help.
Make sure you have everything that smells like food out of your tent, such as toothpaste, soap, hand lotion, and granola bar wrappers.
I don't think there are any standards, but I would shoot for at least 10 feet high, and at least 4 feet away from the trunk of a tree. Make sure the bag is suspended on a branch too small to support the weight of a bear. It should also be hanging down from the branch by around a foot to prevent other smaller animals from having direct access to the bag.
