Sound like a frost proof valve.

Just try pulling on the long pipe/rod with eh faucet washer on the end.

It sounds like the screw holding the washer is wedged into the valve seat.

Is the valve soldered in or is it threaded on. If its threaded then you can unscrew the entire valve.

turn the water back on and stand back. the water pressure will probably shoot the both of them out

Try tapping straight down on it to see if that breaks it loose enough for a screwdriver to remove it.If that doesn't work, if you can't grip it with needle-nose pliers, you might get a small chisel and hammer and try to tap it around counter-clockwise to remove it.

Instead of repairing a frost free, a plumber told me it's just easier to replace the whole thing. Something to do with the clearance to slide out the stem. Note that the new ones at home depot can be a bit wider in one part of the shaft, so you may have to bore the hole out a bit more.