A sling launcher (discussed in this write-up by Landis) is a tower with a motor that spins a hub with two or more cables attached. Payload(s) are attached to the end of one or more cables and counterweights to others. Once the hub is spinning, the cables are reeled out a number of kilometers until the angular momentum at their tips is so high the payload reaches orbit after it is released.
The paper linked to discusses its use on the Moon. I can see why drag makes the system unusable on Earth, but the atmosphere of Mars is only 1% that of Earth. There must be a density below which the atmospheric drag is manageable. Actually, considering the high efficiency of sling launchers as a launch system, and their relative simplicity, perhaps it could be worth dealing with quite a bit of drag, considering the savings on propellant.
Is the atmosphere of Mars too thick to make sling launchers practical?
Edit: Sensibly, several people have mentioned that the cable for this launcher is impossible using current materials, including 2012rcampion's answer, which lays out the calculations that show that. The paper referenced based most calculations on carbon nanotube cables. This technology seems feasible if still distant, so let us assume it would be used in this case. From the paper:
The ultimate tensile strength of fullerene nanotubes is predicted by theory to be well over 100 GPa , with measured values on individual tubes approaching this value. Allowing an engineering factor of 5 (including the added strands for cross-connections, discussed below), the material should allow a working stress of 20 GPa. A mass of one thousand kg at 11.5 gravities (110 m/sec2 ) results in a force on the cable of 112,000 N, as shown in table 1, so the required cross-section of the cable is 0.00389 cm2 (0.389 square millimeters) per ton of end-mass. (More likely, this will be in the form of a number of separate cables which sum to a total cross-sectional area .389 mm2 ). The density of fullerene nanotubes is 1.3 gr/cm3 . The mass of the 50 km cable itself is then about 25 kg, and it is clear that neglecting the mass of the cable itself in the calculation was justifiable. The counterweight cable carries half the mass, but at half the acceleration, and half the length, so the counterweight cable has a total mass of about 13kg. An ultimate strength of 20 GPa may be optimistic for a realistic material. If the working stress is reduced to 10 GPa, the cable cross-section is doubled, and the mass increases to 51 kg and 15 kg for the launch cable and the counterweight respectively