Unless my understanding is way off, metallic hydrogen is more or less an exotic state of hydrogen under incredible pressure (and preferably also low temperatures). My understanding is that this is not a normal state for Hydrogen and the Jupiter example demonstrates that; if hydrogen stayed in a metallic form after it was transformed by circumstances like those found on Jupiter, then there'd be a lot more of it observed even in our solar system.
But, to the question at hand; what toys do you need to mine a gravity well like Jupiter?
Jupiter presents a number of challenges to your mining fleet. The pressures as you go down below the clouds will be incredible and will require tougher and tougher outer shells, like submarines. But, you're still moving through gases, meaning that you don't get the benefit of buoyancy. That means you need some way to stay at a given altitude at high pressures and incredible gravitational forces.
In some ways, Venus is easier. Sure, you're dealing with superheated gases of incredible pressure, but at least the gravity is almost identical to that of earth's.
And of course, you've got a massive electromagnetic field around Jupiter, potentially damaging all your sensitive equipment on board.
But enough of the problems; how would you actually do it?
Well, the single best way I can imagine would be some form of anti-gravity.
Gravity is one (and the most exotic) of four fundamental forces in nature, as we understand them - Electro-magnetism, weak nuclear, strong nuclear and gravity.
Before 1861 we thought there were 5, but we learned to integrate magnetism and electrical charge through the Maxwell equations, and this led to a massive explosion in our technological capabilities. We need that same kind of integration with gravity into electro-magnetic forces so that we can manipulate gravity via forces we already understand and can control to some degree.
Imagine (if you will) the ability to take a nuclear sub (no, not a real sub; the usefulness of submarines as spaceships have already been discussed in detail on this site) style vessel, and using the nuclear energy to generate a massive amount of electricity to power the vessel, but also do two additional things;
1) Generate a protective magnetic field, and
2) Manipulate gravitational forces.
Your new vessel could even launch from Earth by making its relative mass 0, then use more conventional propulsion systems to just glide off the surface of the planet and into space. Conversely, it could use similar techniques to hold station in the depths of Jupiter's clouds. The artificial field reduces its mass to zero, or something so small that it's easily counteracted by conventional thrust. The outer shell is strong enough to withstand the pressure of the gases outside, and the protective magnetic field keeps out cosmic rays in open space, protects the internal equipment (and people) from the damaging effects of Jupiter's magnetic field.
Tie all this to a conventional (?) ion or plasma engine for your conventional thrust, and you have a working spaceship that can operate in open space, gravity wells and high pressure environments.
Of course, being able to manipulate gravity directly may also have another benefit; the speed of light is set to where it is because at that speed, any object that has mass at all now has infinite mass because of the weight of the kinetic energy that the object now possesses. If your gravity manipulation device can reduce the apparent mass of an object that also factors in the mass of energy, then it's entirely possible this could be the first step in developing some form of FTL drive.
What all this means in effect is that in order for humans to safely harvest anything from the mantle of Jupiter, we're either going to need advents in material science that are currently impossible to conceive, or we're going to need some form of Grand Unified Theory (GUT) that not only makes travel within Jupiter's gravity well possible, but could also possibly open up the door to FTL travel.
So, it's going to take a while before we can do it.
