Would it be possible to skydive from LEO using current technology and disregarding any financial constraints? The jumper’s equipment would have to meet the following qualifications—it has to be a jet pack. Any sort of external, enclosing structure would be cheating, even just a scaffolding skeleton to which required equipment, such as spare fuel tanks, etc. would be attached. The jet pack would be attached to the diver using a typical shoulder harness, and the bottom may extend no lower than the jumper’s kneecaps. Since money is no object, we can dispense with hydrogen/oxygen fuels, and entertain the possibility of exotic fuels which have a higher exhaust velocity such as fluorine-based propellants, if such a thing is even possible. I’m guessing that building a spacesuit capable of surviving re-entry temps (and thus ensuring the survival of the jumper) would be a very ambitious, but doable engineering feat. It doesn’t matter how thick the suit or the jetpack are. The jumper could look like the Michelin Man met the Hulk. Could it be done?
People have dived from lunar orbit and landed just fine. The problem is that there were inside a capsule and even if you get the materials for a suit which could withstand reentry temperatures, I suspect the speeds are such (11.7 Km/s from Lunar orbit and 8 from LEO IIRC) that the individuals would be ripped to shreds.
Its already a problem ejecting at high speeds from aircraft.
You have to remember that high speed re-entry heating is ***not *** primarily due to the high altitude, it is because of the high delta-V (or horizontal speed) required to reach and maintain an orbit.
Felix Baumgartner recently did mostly what you’re describing and he did not have to be concerned with re-entry heating because he had very little to no delta-V. He merely jumped from a floating balloon. Because the air was so thin at that high an altitude he did actually fall supersonic briefly, but he experienced minimal heating as the atmosphere gradually thickened and he gradually slowed down. His biggest concern was not entering into an uncontrollable flat spin and passing out or even dying from hypoxia (lack of blood flow and oxygen to the brain).
If we could build a highly efficient rocket engine that could carry enough fuel and was powerful enough to fly straight up to an altitude of several miles (even higher than Baumgartner’s balloon) then again, re-entry heating would not be a major concern for a skydiver jumping from it. They would have to be concerned about the rocket hitting them as it would also be falling straight down along side them*!*
The so-called ‘jet pack’ you’re describing would have little to do with countering the effects of falling from gravity. It would essentially have to serve as a retrorocket pack, negating the astronaut’s high orbital speed (delta-V) in order to begin descending toward the Earth’s surface. But you could not ‘stop’ so fast as to avoid slowly entering the atmosphere at high speed and burning up.
We have several of those already. Both the Blue Origin New Shepherd and Virgin Galactic SpaceShip 2 effectively do that, just go straight up and back down again, in New Shepherds case to 103 km last test flight. So you’d “just” have to figure out a way of opening a hatch and letting the sky diver out at the apex.
The simple answer is “no”.
As has been noted above - just getting up to the same altitude as LEO is not the same as being in LEO. Not by a very large difference in velocity, and hence energy.If you were a passenger in a souped up version of Virgin Galactic’s craft - one that could make say 200km up, and you simply stepped out of the craft when it reached the top of its trajectory, you would need somehting not a great deal more sophisticated than Alex’s suit. More air, more thermal management, about it. (I can see this becoming a big hit in a few decades time - offering space jumps to wealthy thriil seekers - heck I’d do it.)
But of you are actually in LEO, you have a huge amount of speed you need to get rid of. To get rid of that speed with a rocket motor you will need a rocket of approximately the same size as one that can give you that speed. Ie one that could put you into orbit from the ground. Now one launching fro the ground suffers a few issues you avoid in space - it has to fight atmospheric darg for a short while, and it operates at atmopheric pressure, so the engines are not quite as efficient as in vacuum, but these are only a matter of percentages. No matter how exotic your fuels (again, over liquid Oxygen/Hydrogen you are only getting percentage improvements, not anything dramatic) you are never going to get a backpack device that can deliver the delta-velocity you need. Even just getting a guy in a spacesuit slow enough is going to be something the size of a bus, not a backpack.
So you are left with the more conventional way of getting back. Wash off the energy in the atmosphere. Use a small retro-pack to get you down enough to start to get some atmospheric drag and let it go from there. So you don’t need a jet-pack - you need a heat shield. Which isn’t all that hard. But the OP specifies that it has to be a back pack, which isn’t exactly the most easy constraint. Perhaps if the astronaut curls up in the foetal position and goes in backwards you could make something that worked. I would not however be first in line to try it out. Something more akin to a large surfboard (as used by Lt. Doolittle) might be a better bet 
This reminds me of the high school entry question about how fast dropping a tennis ball from a moving car.
“Just stepping off” doesn’t mean “straight down” from either a ballon or a LEO geosynchronous. As the world/atmosphere turns, you would still want the most efficient heat loading time envelope for a given deflector.
There is no “straight” (vertical) descent without infinitesimal (literally, in theory) accelerations, even if true normal to the globe descent is a good thing (I have no clue), although the effect is minimized with Baumgartner–the elevator-shaft going down effect, or Road Runner over a cliff after the “gulp.”
IIRC there was some sort of emergency orbital to earth escape ‘ball’ planed. Basically a large inflatable ball that one climbs into that somehow get it to reenter the atmosphere. While not technically qualified under the OP’s conditions it would be the closest thing to that.
On Edit, this is what I was thinking of MOOSE - Wikipedia
Coming down from the height of the moon, I suspect an astronaut would asphyxiate to death (because his suit would burn through due to Mach heating) before mechanical injury became a problem.
The bodies of the astronauts in the Columbia disaster were apparently discovered in a dismembered state, but I wonder whether this might have been because the crew cabin resisted breakup (and expulsion of the astronauts) until it had reached an altitude where aerodynamic effects at the speed of its breakup were sufficient to dismember their bodies. The aforementioned body parts had also been charred; not clear whether this happened before or after the dismemberment, though a forensic examination would have determined the sequence by looking for charring on tissues exposed by the dismemberment (maybe this info is in the CAIB report?).
Remember not to confuse MOOSE (which was never developed much) with the PRE (Personal Rescue Enclosure) which was basically just an inflated ball the astronaut could curl up in and be carried thru space to a rescue Shuttle. Except for the first four ‘qualifying’ launches, the astronauts didn’t wear space suits on the Shuttle (until after Challenger).
Great cite/site.