Why would gas leak from small hose on bottom of carburetor float

Question edited for clarity.

I have ridden an XT500 on Green Lanes in the UK. Great old thumper!

You will probably find there is a small overflow hose from the float bowl of the carburetor. The float valve is sticking. Remove the carb and clean all the jets, especially the one in the bottom float bowl.

https://www.google.com/search?q=1979+yamaha+xt500+clean+carburetor

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I think you may have a stuck float in the carburetor.

Your carburetor float valve may be stuck open. You'll have to remove the carburetor, remove the sediment bowl at the bottom of the carburetor (where the float is), and check the float. Is it filled with gas? Replace the float if it is. If the float is dry inside, then remove the float and clean the needle valve & seat. (be careful to note how the float pin, linkage, spring, etc. are positioned so you properly reassemble. After cleaning & reassembling, turn the carburetor upside down & try blowing into the gas line connection on the carburetor. You shouldn't be able to blow air. Turn the carburetor right side up & repeat. If you can blow air into the carburetor's gas line connection, the float valve & seat are properly working. Reinstall the carburetor. You should be good to go.

The carburetor over flow tubes should be open. Here are the possibilities. 1)Your carburetors are dirty and need to be cleaned. Clean your carburetors. 2)Your carburetor float needles are damaged and causing the carburetors to over flow. Replace your float needles. 3) The fuel filter is missing on the fuel petcock and rust is causing the carburetors to overflow gas. Put the fuel filter back over the carburetor inlet tube and/or add an inline filter to stop the rust from sticking the needles open and over flowing the carburetors. 4) Your floats have become saturated with fuel and no longer float in gasoline. Replace your floats. Poor idling or idling on only one cylinder can be caused by a bad spark plug cap. If you unscrew the spark plug caps and OHM them out they should read 5000 OHMS no higher and no lower. Good luck you have your homework.

Hello Pete,

Is fuel actually leaking from the bottom of the bowl, or is leaking out the air filter and running down to the bowl?

If the former, then your bowl bolt's gasket has likely deteriorated. This would allow fuel to run down the bolt itself. Note that some carburetors will have a fuel shutoff solenoid (to the engine, not to the carburetor) that holds the bowl on, rather than a bolt. There will still be a gasket though. Turn fuel off before trying to service.

If fuel is running out from inside the carburetor (through air cleaner, etc), then you have a float / needle that is stuck. This is quite common after a small engine has been in storage and fuel was left in the carburetor. The residue causes the float to stick to the bottom. Turn the fuel off, undo the bolt that holds the bowl on, and carefully remove the carburetor bowl. Use a can of carburetor and choke cleaner to spray out the "gunk" in the bowl, spray off the float, and then reassemble. If it still leaks, then your needle seat will need to be replaced as well.

Remove the air filter and set aside.
Take a can of carburetor cleaner and spray the inside of the carburetor.
Let it set for a moment and then try starting the engine. It should run a few seconds. Repeat. When it starts squirt a small shot of cleaner in the carburetor to keep it running. That should remove the varnish the gas leaves as it evaporates over the winter.

If it still won't keep running then there is trash in the float bowl plugging the needle valve seat and the jets.

1) Turn off the gas or remove the gas line and plug it with a small screw or stick to keep it from leaking out of the gas tank.
2) Look on the bottom of the bowl for a nut .
3) Remove the nut and drop the float bowl.
4) Carefully remove the pin holding the float and needle valve assembly.
5) Take the carburetor cleaner with the straw it comes with and spray every hole in the carburetor to clean the jets . (up through the needle valve seat, down top of the carburetor , see a hole, spray it with the cleaner.)
6) Clean the float bowl.
7) Reassemble and make sure the gas isn't leaking and try to start it again.

Carburetors have a Float Bowl on the bottom. This float bowl has a small amount of fuel, that the carburetor uses. (The float bowl's amount of fuel, is replenished from the gas tank. The fuel is gravity fed from the gas tank, through the Fuel Line)

The float bowl has a Float Needle inside. The float needle is attached to the Float. The Float resembles a thick doughnut. On one end of this doughnut, is a flat piece of metal shaped like a U on the end. Looking at this -> U shape, there are two holes at the top. There is a metal pin that goes through these holes, and the pin is attached on it's end's, by the main body of the carburetor.

The flat metal U shape, and the metal pin, works together to form a type of hinge, allowing the float to pivot on it.
Where the U shaped hinge is attached, the float needle, clips to it.

The float needle has a tapered neoprene tip, and sits in a Float Needle Seat. This seat is a barrel shaped piece of brass, and has a tapered hole at the bottom. Fuel comes up through this seat, and the float needle regulates how much fuel can come in.

As the float drops down, it draws the float needle out of the float needle seat, and fuel comes in. As the float bowl fills with fuel, the float rises, and pushes the float back up, also pushing the float needle back into the float needle seat, cutting off the flow of fuel.

There is a specified height of the float, in relation to where the float sits in the float bowl. If the float sits too high, the float bowl fills up with fuel, and overflows. This overflow goes through an Overflow Tube, on the carburetor.
(If the float sits too low, the engine starves for gas)

A LOT of times, crud builds up in the gas tank, makes it's way through the fuel line, and gets into the float bowl. It can get in-between the float needle, and the float seat, thereby flooding the carburetor out, and fuel comes out of the overflow tube.

The float bowl is also attached to the carburetor with a brass main jet nut. At the bottom of the float bowl, this will look like a brass head of a bolt. Under the brass head is a small O-ring. This can leak. First step is too insure this is tight. (Do Not Over tighten! This is part of the main jet, and is expensive! It's Brass)

Where the outer part of the float bowl itself, attaches to the carburetor, there is another O-ring.
A large one. This O-ring, like the main jet O-ring, can deteriorate over time. Additives that gasoline has in it, heat from the engine, and heat from the sun does this. Plus just age. Neoprene breaks down after time.

It could also be that the neoprene tip of the float needle has broken down.

You indicated you adjusted screws on the bottom.
These are the Main Jet, and Idle Jet -> Air Mixture screws. All they adjust is how much air goes through these jets.
The amount of fuel going through, is predetermined by specific passageways, that are drilled through the main body of the carburetor.

This only comes into play when the engine is running. With the engine off, and fuel leaking out, you have crud under the float needle, or one or more of those O-rings has deteriorated.