One of the best things about hammocks is that you can elevate them as high as anchor points allow. That means you can sleep with the woodpeckers and squirrels up high. It's actually a lot of fun to do, but it isn't for the faint of heart, or those who sleepwalk. Getting up high among the branches lets you hang all sorts of stuff and have an arboreal campsite, it's especially good to use where you have concerns about the ground...swampy areas, gators, snakes, rabid coyotes, meth addicts stumbling around in the woods.
If you hang your gear away from your hammock and haven't cooked and eaten right underneath, you should be fine. And yes, bears can climb trees. They aren't ninjas though, and if you aren't awakened by the sound of them clawing their way up a tree a few feet away, you might as well sleep on the ground and not worry anyway. If one does go through the trouble of climbing a tree after you wake up and start yelling at it and shine a light on it, you still have recourse to a sidearm or using a pre-placed bailout rope to swing away like Tarzan. Makes for awesome selfies too.
I usually prefer tarp/fly shelters and sleeping on the ground, and have done plenty of that while hunting in bear country. In bear country just try to give yourself a protected space...like setting up in blowdowns with trees/logs piled on three sides. And then you have built in alert/protection (noise to climb over, barrier) in three directions and one approach to worry about. Pile brush to narrow it, and build your fire there. People have used that technique for millennia.
Really, don't worry. Just sleep with your pistol and knife. All smart bear tacos are armed tacos.