I am rewiring my master bedroom. I bought 14 gauge wire with a black, white, and copper wire, which is the ground. The ground wire has no rubber on it. When I have all the wires going into a junction box, the ground wire touches the side of the box. Is this safe? Is the ground wire suppose to have protective rubber around it?

The Box should also have a ground wire. I like to make them up first, tuck them in the back of the box, then the neutrals, last, the hots and switchlegs, if in switchboxes.
All of the grounds are connected to the box/same potential.

If it is a junction box, there should be a ground clip or pigtail to a ground screw( I'll take a guess we are talking about a metal box) Where is the junction box? There is no covering on a ground wire and touching the box is fine. If you are calling a wall box a junction box then you would use a ground clip or screw if a metal box, and just connect the ground to the green lug on the outlet or switch. ONLY 1 WIRE per screw. If the box is along the wall and in parallel, use a pigtail lead for the ground. Good idea to wrap the outlet with electrical tape and cover the terminals. Depending on how much rewiring you are doing, your local code may require an Arc Fault breaker for the bedroom circuit. That is a requirement for all new homes.

Agreed but arc fauly is not required by a local code. International building codes specify arc faults for bedrooms.

I just built 2 new homes in Georgia and Arc Faults were required