Would you die if you put your hands on a powerline?
Hanging from a power line you should be as safe as a bird.
The voltage difference is between the lines (e.g. in a 3-phase system) and between the line and ground. This voltage difference exists across the insulators and pole, as well as through the air to ground. These voltage differences are obviously small enough to avoid striking an arc, hence no current flows between the lines or between line and ground. If you are hanging from one line, there is no change in the separation between the lines (unless you are swinging wildly) and hence again no current flows between the lines. As the distance between the lines will usually be smaller than that between your feet and the ground, again no current will flow, and you will be safe. Note that, if this distance were too small, you would not be safe standing under the line either!
Your real problem will be to get down from the line. Unless someone can switch off the power, or you are an acrobat who can jump from the lines to the pole, you will need to touch both the line and the pole simultaneously. If you are hanging from high voltage lines (tens of thousands, not hundreds), touching even a wood pole at the same time as the power line may kill you, unless the wood is extremely dry. Even though wood is considered a poor conductor, when it gets damp its conductivity increases dramatically.
Have you seen the helicopter crews that work on overhead lines approach a live wire. They extend a conducting pole and equalize their voltage with the line. After than it is "safe" to work on the line. As long as there is a conductor connecting them to the wire it is ok. Of course the initial equalization process would kill you if the arc went through your heart instead of the conducting pole.
YouTube video of what I am talking about is seen here.
So in theory, yes it would be safe, if conditions are ideal, and everything else stays way away from you, so you don't arc to a nearby tree or something.
PS. It is never "safe", just "safer".