Earth magnetic field space elevator. No cable

Setting aside the technical details, the biggest problem with this idea would be Earnshaw's theorem, stating that it is impossible to stably levitate a magnetic dipole in another magnetic dipole's (Earth's) field. In other words, you could theoretically generate enough magnetic force for lift-off with your dipole (the coil), but you would have trouble stabilizing it so that it doesn't topple over and fly back into the Earth.

Edit, after bringing up active stabilization in the comments:
While in theory, yes you could actively stabilize the whole setup. And just for the sake of arguing, you could also stabilize the ring by (mechanically) spinning it around its axis. The torque that would act to topple over the dipole would now cause precession instead. Voila: the ring is stabilized.

However, I still don't think that this whole idea would work in reality.
The magnetic moment of the dipole is $m_{B}=IS$, where $I$ is the current through the loop and $S=\pi R^2$ is the enclosed area by the loop of radius $R$. Since we're talking about a setup on the magnetic axis of the earth, the geomagnetic field can be written as $B_{\rm E}=B_0(1+h/R_{\rm E})^{-3}$, where $B_{0}\approx62\,$µT is the field strength at the surface of the Earth, $R_{\rm E}\approx6400\,$km is the radius of the Earth, and $h$ is the height above ground. The force generated by our dipole lifter would be: $$ F=-\frac{\rm d}{{\rm d}h}\left[m_{B}B_{\rm E}\right]\ =-\pi IR^2B_{0}\frac{\rm d}{{\rm d}h}\left[(1+h/R_{\rm E})^{-3}\right] =\frac{3\pi IR^2B_{0}}{R_{\rm E}}(1+h/R_{\rm E})^{-4}, $$ We can also approximate that $h\ll R_{\rm E}$, so that $F\approx3\pi IR^2B_{0}/R_{\rm E}$.

Now, then lets say that you build yourself a ring with radius $R=500\,$m, and drive a current of $I=10\,$MA through it. That would generate a lifting force of about $230\,$N (newton), or about $23\,$kg ($50\,$lbs for the imperial entanglementalists).

But how much would such a device weigh?
We, of course, use superconductors in order to not require extremely thick copper or silver conductors. But the problems with superconductors is that you can't just drive however large current you want through them. At some critical current density, the supercondictivity will break down due to the high magnetic fields generated by them. Lets say we build that loop of niobium-nin, a superconducting material with a reportedly record-breaking current density. It's highest current density is $j_{\rm max}\approx2600\,$A/mm$^{2}$.

We wanted to push $I=10\,$MA through it, so that would require a cross-sectional area of $I/j_{\rm max}\approx4000\,{\rm mm}^{2}=0.004\,{\rm m}^{2}$. If we multiply that by the length of conductor needed, $L=2\pi R\approx3000\,$m, we get a volume of $12\,{\rm m}^3$. I don't have the (mass) density of niobium-tin, but needless to say $12\,{\rm m}^3$ of it will weigh in the order of at least $20{-}100\,$tons, and that's not counting all the cooling apparatus you would need in addition to the pure weight of the superconductors. (And I haven't even gotten to address the mechanical strength required for you to spin the whole ring at hundreds or thousands of rpm, in order to stabilize it.)

Tl;dr
You won't be able to generate enough force to even remotely lift your apparatus. I hope this can make you sleep well tonight.

Here are some problems that I see :

Scaling issue mentioned by earlier answers, primarily limited by the current density allowed in the superconductor.

The superconductor needs to be high-temperature. Yes, even in the Arctic, it's too warm for known superconductors in 2019. The superconductor needs to operate at low pressure (several have been found at very high pressures, but this is in the upper atmosphere and you cannot afford to bring giant compressors with you). The superconductor needs to be resistant to or protected from atmospheric oxygen, cosmic rays, the Van Allen radiation belts, and other standard problems of rocketry.

Weakness of the Earth's magnetic field. This could be worked around by building a HUGE high-powered ring on the Earth, aligned with the Earth's normal magnetic field, and a smaller reversed ring to fly into space.

Weakness of most ferrous metals and other conductors to induction currents from the rings. The spacecraft is crushed and torn apart.

Finally : Orbit is not a matter of altitude, but a matter of speed. So your spacecraft must take significant rockets along with it. Lofting to a mere 35,000 km , the standard geosync altitude, would require a LARGE amount of delta-V that has to be applied promptly. Trying to go higher would require less rocket delta-V, but your magnetic lifter works worse and worse the higher you go.