2001 volvo S60 door would not unlock with key

Do you mean the remote doesn't work (check battery) or the key in the door doesn't lock/unlock it (check the linkage in the door between lock and electronic lock mechanism)?

one of two things are going on.
1. The door lock cylinder is worn out needs replacing.
2. It's not recognising the key fob. Try this..
you can use your key to open the trunk ,thankfully! , try and unplug the battery, and then connect the two wires that go to the battery together for about a minute, and then reconnect them both to the battery. It's call a hard restart, and it might fix your problem.
won't mess anything up by doing this.
Hope this helps a little.

Use the key and manually lock and unlock the drivers door.

Inside the truck behind the trim is a manual release for the gas door, the release looks like the head of a large key on a small wire string. It is marked.

With key in hand,hit power door lock switch to lock. Raise release handle & lower. Hit power door lock unlock raise & lower handle. May take a couple times is stuck in limbo between lock & unlock mode.

yes it is in trunk, open back door and find out one pull knowb exactly back side of fuel lid door. just pull and door will open.

I, feel sorry for you, after reading your problem, but don't worry you visit the Reliable-Store they will help you for sure. Volvo S60 2001 2009 Workshop Service Repair Manual

sounds as if the locking mechanism is wearing out or not making the proper connection to the steering wheel

I had a similar problem but mine used to unlock all the doors. My service people could not solve the problem. Finally I found & solved it myself. I found that the micro switch in the remote control was staying down and not coming back up. I solved it by releasing the pressure on the switch by undoing the screw on the remote.