I know it's not possible with Earth today, but With today's level of material science technology, would it be possible to make cable strong and light enough to make a space elevator system connecting a Mars-synchronous satellite to an anchor on Mars' surface, ignoring any load at all by an elevator car? Or can someone say with a high degree of certainty that the level of strength to weight possible in cables today is still far from possible even with Mars' lower mass compared to Earth?
the simple answer is no. Materials for an Earth elevator are at least one order of strength too weak at this time. Mars gravity is around 0.378 of earth, so materials are still too weak.
The long answer is much more complicated: 1. Taper of the tether plays a role as much as the safety factor you want to engineer into your elevator, how much a tether can hold, and how much material you can put into space to construct it. Strength, taper, and tether mass are related. You can check for the Space Elevator feasibility condition via isec.org or Google. It is basically some form of decay equation: how much do you need to keep lifting to maintain or grow your elevator, basically taking away from transport payload for maintenance/repairs. 2. Mars has some interesting options with Deimos being made of mostly Carbon. Material for a tether could be refined on site, e.g., when Planetary Resources engineers related technology for asteroids. Deimos' orbital period is with 30.30h is close to the sidereal rotation period of Mars with 24.6229 h. You could work a tether that drops from Deimos part way into the Mars atmosphere and get something that you could attach to with much smaller Delta V than getting to orbit, thus limiting tether strength and mass requirements substantially further. I strongly believe that that the Deimos model is probably close to being within reach of today's advanced materials. Of course you need to work out some control to avoid Phobos dropping by every so often below Deimos. 3. The best anchor point for a Mars Elevator, once material becomes sufficently strong for surface attachment would be on Olympus mons. That puts it above sand storms and other Mars weather.
Finally: I believe that Mars is destined to become the shipyard for the solar system. It is sufficiently friendly for construction sites and the lower gravity makes it much more amenable to lift things to space than Earth.
Hope this helps. Let me know if you have any more questions about the material strength.
It is actually an old ladies myth that the space elevator cannot be built with current technology. It would just be extraordinarily more expensive with current materials.
The elevator needs to be built from geostationary orbit downward. This means that all the material needs to be pulled up by conventional propulsion, this is the principal factor affecting cost. The conventional approach to build an elevator without materials satisfying the strong tensile requirements that are required for the single-tether design would have to be an exponential structure; a single tether connected to earth, supported by two tether, supported themselves by 4 tethers,.. until reaching above geo orbit, where we'll have to attach $2^n$ tethers to the counterweight, where $n$ is given by the tensile properties of the chosen material.
If you have already one space elevator, then it is easier/cheaper to build a second one, because the expensive step of moving the mass up has already been bootstrapped. In the case of mars, it would make more economical sense to actually send the building materials from earth than to uplift them from mars surface, assuming of course we already have a working elevator on earth