Is the blank going through the headstock and lathe?
Yes. The blank goes through the headstock.
Do you take any precautions or have any support on the other end?
Yes I do. My support is a simple 2x2 with a 3/4 inch hole in it at the same height above the floor as the hole in the head stock of the lathe. Has worked well for several years. Masking tape protects the blank. Keep it simple.
If you are running it as I think you are, what is you chuck grabbing on to?
This is a bamboo rod that has a layer of masking tape under the chuck.
I've been turning on my Shopsmith
I don't remember if my old Shopsmith had a hole through it or not. I don't think it did. You might have to do things a little different than I do.
The following assumes a hole through the head stock.
If you are making graphite/glass rods I suggest you turn a short dowel and glue it into the butt of the rod. Then, chuck up the rod and center drill the dowel to accept the point of a live center. When I do graphite/glass. and I don't do them very often, I use a 5C collet chuck and a collet to grip the rod blank that has a layer or two of masking tape to protect the blank. The collet grips around 360 degrees instead of just three points for the regular chuck.