Compression fitting are your friend! More on that in a moment.

Depending on the type of faucet you may have worn valve seat or ring or washer. But you sound like you have the same problem I had last summer. One of my toilets starting running all the time because the float wasn't turning off the water once the tank filled. So I was going to replace the guts of the toilet and I started by (well actually I rigged the toilet for the night because everything was closed and I needed to flush it so I used a piece of wood to stop the float) turning off the water supply with the shut off valve. It did the same thing that happened to your hot water shut off valve, I closed it yet it still didn't stop the flow of water to the toilet. So I had to shut off the water at the point it entered the house. I also had the option of doing it at the meter. Then I measured the length of copper pipe coming out of the wall and I came to the conclusion to cut out the old shut off valve and replace it was going to require precision work as I had very little pipe to work with. What the hell though I gave it a try. The original shut off valve was soldered in and I didn't want to go that route so I choose a compression fitting. I have worked with them before so I knew how handy they are. They use a compression nut and compression ring to make a water tight seal. I was worried because the compression fittings I used before were for PVC and metal is harder to make watertight seals. If you have a copper supply pipe, here's how.

So I had limited room to work and bought a tight area pipe cutter which didn't work. So I used a hacksaw and cut the old valve out right behind the valve. I cut it so close that I actually cut a small bit out of the valve which was hard to do as those valves have thick metal. I then cut the pipe again removing the thinnest little ring of metal, thinner than a dime, to make the cut straight and true from top to bottom. That's step one.

Step two use an emory cloth to clean the pipe and remove any rough burrs on the pipe. Then slide the compression nut over the supply pipe. I used super fine sandpaper.

Step three, place the compression ring over the end of the supply pipe.

Step four, thread the compression valve into the compression nut. This is the hardest part or was for me as you want it to seat evenly (thread itself on right). Hand tighten only at this point because you can overcompress it and then it won't work or if it isn't seated properly you will strip out the threads. When you have it on right the you tighten the valve and the nut with 2 wrenchs, an opened ended wrench is preferred so as not to strip the outside of the valve. Try not to use pliers or a vise grip. One wrench to hold the valve in place the other to tighten the nut. You want it secure but not overtight.

Now sadly here is where we may have to depart. The toilet supply line and the shut off valve were easy to connect as I was able to use a braided steel line with a nut that connects right onto the valve, easy as anything. Then the other end connects to the toilet and that was easy too. Both basically ends basically screwed on. For your hot water supply line to the faucet, there may be many different ways it connects and it could be an easy fix to a hard fix at that point. So look at how the shutoff valve connects to the faucet and go to one of those do it yourself warehouses and ask for some help tomorrow. Take pics if you have to. They may say oh wow here's all you have to do or they may say yeah you're screwed.

If you can do all this then you can fix the seat or the washer or ring in the faucet as that is much easier. Hell you may start there and not have to fix the leaking shutoff valve. Well when I was done the toilet worked better than ever as I used "guts" that were much more quiet than the original, used that steel braided hose which is much more safe against sudden leaks, and a shut off valve that only requires a 1/4 turn to close the water supply. In the end I hadn't broken the valve but a piece of metal got into it so when I closed it the metal was keeping it from closing all the way and thus not shutting off the water supply. I mave have been able to fix it by flushing it but like I said that repair is better than original and it was a good feeling knowing I did a job well done and it looks good too.

So determine how the supply line connects to the faucet and the shut off valve and if you can reconnect the supply line at both ends. This is probably the do it yourself limiting step in the process. Then see how much room you have to work with under there and how much pipe you have to work with. If you are handy you may be able to do all of this. If your DVD player is blinking 12:00 then you might not.