Could you have a large winch in space with a cable hanging down, say, 60 kilometers, connected to a floating station floated by helium? The payload could be floated up by balloons, which would then dock with the floating station. The payload would then be attached to the space-winch, which would bring it up to space. Would that be possible, or is there a gaping hole in my logic?
Is your objective to "get to space" or to "get to orbit"? The two are fairly different - just "getting to space" is not that hard (space starts at about 100 km up), but in order to stay there (ie be in orbit around the Earth) you need to be going sideways very very fast (about 8 km/s) - What If has a nice visual explanation of this.
So assuming you want to get to orbit using your method. You therefore need a a satellite in low Earth orbit (say, at an altitude of 200 km - if you're much lower than about 160 km, the drag from the upper atmosphere will slow your satellite down too much and it will re-enter the atmosphere), and you have a 200 km cable hanging down from the satellite. At an altitude of 200 km, the satellite will orbit the Earth about once every 90 minutes (that's how fast it has to go to stay in orbit). Assuming an equatorial orbit, the bottom end of the cable would be dragged through the atmosphere much much faster than the Earth is spinning (the Earth spins once around its axis every 24 hours); since it has to go around the Earth once every 90 minutes, it will be going at about 26,000 km/h (7.4 km/s). Needless to say, not only would the cable burn up almost instantly at that speed, but you would have great difficultly in "catching up to it" in order to attach your payload at the bottom end! And even if you could manage those two, the drag from the cable being pulled through the atmosphere would slow the satellite down so much that it would re-enter the atmosphere in short order.
If you move your satellite into a higher orbit, the speed of the bottom end of the cable would decrease (as satellites in higher orbits take longer to go around the Earth). Eventually, once your satellite is at an altitude of about 35,800 km, it will be going around the Earth once every 24 hours (this is called a "geosynchronous orbit"). However, by that point your "winch" cable will be very long indeed, and the forces on it will be enormous. So enormous, in fact, that no currently known materials can withstand them, although there are several experimental materials (such as carbon nanotubes) which could, in theory, be used. This concept is called a space elevator, and while the principle is moderately well understood, it's beyond our current technology to build one on Earth.
(Note that I've oversimplified greatly - a real space elevator would need to extend well beyond geosynchronous orbit to remain stable (as its mass is now non-trivial, and the material needs to stay in tension), so the length of your "winch" would be even longer.)