Drill through a broomstick handle
VERY carefully?

It would be almost impossible to drill all the way through a broomstick with a normal drill, even if you find one long enough, or use enough extensions. A drill that long will be quite flexible, and will tend to drift with variations in density of the wood. The grain will hurt you.

What does work is to work essentially backwards. Drill a long hole through a wide enough piece of wood that the drill will still come out somewhere near where you want it to do so. Then mount the blank on a mandrel, and turn down the wood into a barrel. You can get an amazingly thin result by this method. Note that while the mandrel will provide some stiffening, it is often helpful to have an additional support to steady the blank, preventing "whipping". This is called a center steady, or a steady rest. I've turned long poles the diameter of a broomstick up to the length of my lathe bed, so about 3 feet long, but this takes care to do well.

Note that you still need to drill through the original block of wood. So a drill press of some sort is often a good choice here to drill holes that need to be perfectly aligned with an axis. If you have a good lathe, it will sometimes have a tailstock that will allow you to drill through the tailstock. This helps to keep the drill aligned with the axis of the blank. But even there, a very deep hole is problematical.

And of course, any time you have such a deep hole, you MUST make sure the chips are allowed to clear out of the hole you are drilling. This means you need to back out the drill often to clear them out.