The water supply to my toilet is copper with a valve sweated on. The line from the valve to the tank seems to be a compression fitting. The valve has developed a leak around the shaft which the handle is mounted on. There seems to be a "nut" at the base of the shaft which looks like I could loosen it and repack the valve but I'm unsure. I'd really prefer to avoid having a plumber come out to sweat on a new valve if I can.
I have repacked these valves many times using Teflon rope. I have even done it with Teflon tape but the rope is easier and last longer. Turn off the main supply and bleed the pressure remove the gland nut and back the assembly out (good time to inspect the seat and replace if deformed), I have a kit of seals I purchased years ago with 4 or 5 different sized seats, some shaft packing (Teflon rope) and some new rubber seals a good idea for any home owner to have a kit like this to repair faucets under 10$ and just add as you use up and save over a replacement. I will usually wrap the shaft with at least 3 full turns and more if the shaft is in bad shape put the assembly back together and tighten the gland nut turn the water back on bleed any air out of the line and check for leaks. I do not back seat (fully force the valve open as this is hard on the packing) I learned this with SCUBA diving where you open the valve fully then close it 1/2 to 1 turn to extend the packing life.
Those valves (unless very old) do not have traditional packing in what would still be called the packing gland under the nut. They use tapered/beveled rubber gaskets there instead. Not really designed to be serviceable I'm afraid, more like disposable.
You can often stop the packing leak by simply back-seating the valve; that is, opening it fully till it stops. That brings the backside of the valve seat back against the valve stem hole at the bottom of the packing gland. Also try carefully tightening the packing nut if you haven't done that already.
There is a chance you could find a similar replacement rubber washer if you bring it to a good plumbing shop. You could try to use some graphite packing string, who knows, it might work...