In theory I believe that if you moved far enough above the surface of the pole you should be able to reach an area where you have constant sunlight, even in the middle of winter.
My question is - how far up would you need to go? Would it be possible to do this on earth and still be able to breathe?
Yes, this is possible on Earth, but you will not be able to breathe.
The Earth's axial tilt of 23.5 degrees means that at winter solstice, the pole will be 23.5 degrees tilted away from the solar terminator. This converts to a distance of $$\frac{23.5}{360}*40075\text{ km}=2616\text{ km}$$
Trigonometry can be used to calculate the minimum height one has to be to see the horizon from a given distance.
To see the horizon 2616km away and therefore become sunlit during the winter solstice, you need to be at the height of $$\biggl(\frac{1}{\cos(23.5^\circ)}-1\biggr)(6317\text{ km})=571\text{ km}$$ above the Earth's surface.
This is in outer space, at the orbit level of satellites.
Because of the tilt of the earth's axis:
In Svalbard, Norway, the northernmost inhabited region of Europe, there is no sunset from approximately 19 April to 23 August. The extreme sites are the poles where the sun can be continuously visible for a half year.
Of course, if the earth's axis of rotation was precisely perpendicular to the direction light travels from the sun, at the poles you could see the sun constantly all year round.
As the earth's radius is about 6356km and its axial tilt is about 23.4° from vertical, you would need to be about 569km directly above the pole if you want sunlight at all hours, all year round. That would put you a bit higher than the International Space Station, so you won't be able to breathe and you probably can't build a tower this tall. For comparison, Mount Everest is about 9km tall and people die climbing it all the time. However, you'd be about 30,000km below geostationary communication satellites and about 360,000km below the moon's orbit so although you'd be high up, in space terms you can go a lot higher.
As you can't build a tower that tall, if you don't want to fall back down I'd recommend being in some sort of orbit.
Needless to say, if you're trying to avoid being too sci-fi and you can tolerate the constant days only lasting 4 months or so, a visit to Svalbard would involve a lot less space travel.
You might try a sun-synchronous orbit. Because the Earth is not a perfect sphere, satellites' orbits move over time. By cleverly designing the orbit, you can get the plane of the orbit to rotate once a year, so that the orbit is always in the same alignment with respect to the Sun.
Earth-observing satellites love to use this type of orbit. They can pass over the day side of the planet close to local noon on every orbit (and local midnight on the night side). However, turn this orbit 90 degrees, and you have a terminator-riding orbit. That is, the orbit always passes over the the line separating the sunlit side of Earth from the night side. Such an orbit would have an inclination of 23.4 degrees, and an altitude of around 3100 km. It would still be affected by eclipses though.
Note that geostationary satellites also enjoy constant sunlight. Since they are so far away from the Earth, their 23.4 degree inclination from the ecliptic takes them far above and below the Earth's shadow (except around the equinoxes, when their orbit crosses the ecliptic right behind the Earth).
Well, to make your life simpler, you could just wander from the Artic Circle to the Antarctic Circle and back once a year. If you could accomplish the trip in about 10 hours, you'd be in business; you'd have constant sunlight.
But if you're wedded to the idea of being above the surface of the Earth, and as long as it's been demonstrated that you'd have to be in a Low-Earth-Orbit track / altitude at the pole, why not just fly in an LEO satellite that orbited the Earth above the Great Circle that's perpendicular to the radius of the Earth's orbit around the Sun. People on Earth would be going through night and day as the Earth rotates, but you'd be orbiting around in a circle that formed the base of a very long cone with the Earth at the bottom, and the Sun at its apex. 100% sunlight, 100% of the time.
