Please don't bite my head off. If this is so stupid that it deserves to be squashed, I'm quite happy to delete the question. I mainly would like to know whether anyone's ever suggested this (I searched).
Supposing you have a long piece of string... I mean really long, 500 thousand miles in fact, and you make it into a loop and you attach buckets at intervals along its length.
Then you have a couple of fixed spindles, one on Earth and one on the Moon, and you start pulling on the string: if you wanted to get stuff (people, machines, commodities, etc.) down from the Moon simultaneously with stuff up from the Earth, wouldn't you in fact balance out the effects of the gravity wells (both Earth's and the Moon's)?
I feel sure that someone's going to point out the rather large strains that would be experienced throughout the length of this piece of "string" as the pulling took place*. Naturally I'm suggesting that instead of string it should be made from some suitable 22nd Century technology: maybe as well as being miraculous, the constituent material would also have to expend energy in some way (using solar panels presumably) in order to function.
Compared to the space elevator idea, OK, it's a bit longer. But the space elevator idea faces the challenge that, up to geostationary orbit height, the entire structure has to be supported from beneath, in the very bottom of Earth's gravity well. The engineering specifications of the material of this string would be challenging in a different way.
NB I'm aware that, in practical terms, rockets are in fact a pretty cheap and cheerful solution for getting stuff out of or into Earth's gravity well, particularly if you can make them re-usable. Talk about building a space elevator, space guns or "orbital tethers" etc. still continues though.
Jcaron's comment about the Moon not being geostationary made me think: of course it is primarily the spin of Earth which is the problem here, rather than the orbit of the Moon. This also indicates that, unlike with a space elevator, you would not want to have your maritime tether platform anywhere near the Equator. Instead you would want it to be as close as possible to one of the Poles: my knowledge of the trigonometry involved here is a bit lacking: feasibility would depend on factors like the tilt of Earth, the fact of the Moon's orbit being, unfortunately, inclined 5 degrees relative to the Earth's ecliptic (not equatorial) plane, etc. With the tilt being in the "wrong" position completely relative to the Moon's position once per month, I rather doubt whether you could site the platform actually stationary, AT the North or South Pole.
Instead this platform would probably have to travel at a constant several hundred km per hour, along a latitude line close to that of the Antarctic Circle, where there is less land than with the Artic Circle, doing one circuit per 24 h (length a technically challenging 16,000 km = some 670 km/h!). Although there is little land there, there is the pesky business of ice. Lots of ice. This may disappear in the near future of course.
Another possibility is to station your Earth tether at the South Pole, make it stationary, but disconnect it for maybe half the days in any given month, when a line between the Earth tether point and the Moon would pass through the mass of the Earth ... but ... at that point in the month the North Pole would be workable... so, yes, you have TWO polar stationary tether points, and you switch the Earth end of the loop between them every two weeks - problem solved!
Fortunately humankind has always relished a challenge.
I've done a bit of thinking about this since I posted this. The super crucial thing to bear in mind is that each link must be "smart". As a first hypothesis, each link might be 10 m in length (requiring approx. 80 million of them), and the loop would loop at a speed of 100 m/s. By my calculations, this means it would take about 45 days to transport something to or from the Moon. The "turnstiles" at each tether point might be 1 km in diameter or so.
Each link contains two crucial things: a solar array, which deploys only outside Earth's atmosphere, and a set of gears. The gears are powered by the array. The gears have two functions, without which this space loop could never work.
Firstly, the gears are responsible for driving the loop: throughout the length of the loop, in space, the "up" strand rubs up against the "down" strand, and the gears are therefore responsible for driving the loop mechanically. On this subject, it might be worth wondering what forces would actually conspire to slow the loop once set in motion. Friction between the links? I think the amount of power needed would turn out to be quite minimal in fact, relative to the potential solar power captured along 2 x nearly 400,000 km of links.
Secondly, and more controversially, the gears would be responsible for countering Earth gravity near Earth. At the South/North Pole, where we have our Earth tether point, the loop is stretching off towards the horizon, horizontally. This is determined by the nature of the Moon's orbit and the Earth's axis relative to it, and there is no getting around it. Unlike with a space elevator, we are not exploiting centrifugal force in any way. So a legitimate question is: "why doesn't the loop just fall down?".
The answer is not something to do with tension (some unvarying, dynamic tension might exist, between adjacent links, but nothing like enough to pull the chain "taut": motion of the chain would be the result of the powered gears but above all momentum), but instead that the loop is using the immense amount of electrical power generated constantly by the solar arrays along its length to apply a dynamic "curving force", to "curve" the loop away from the Earth at the most gravitationally difficult point of the loop, i.e. the Earth tether point. This means that, as it heads to or from Earth, in proximity to Earth, at 100 m/s, each link is applying a non-negligible force to its adjacent links, using its gearing, to act against and neutralise the effect of Earth gravity.
At the lunar tether point there are no problems of this kind: firstly, since the loop's attachment to the lunar tether point is vertical, but also because the gravity is much lower.
Since the solar arrays don't deploy in the Earth's atmosphere, you need to get the power transferred from the links currently in space.
Apart from cost, the biggest objection to this idea might be aesthetic: would we really want to look up into the night sky and see an unsightly chain stretched between Earth and Moon?
* Maybe you might also find the Moon is being pulled out of orbit due to the strains involved, but you could always "correct" this by firing off thrusters stationed on the Moon (humanity might want to do this anyway in future, as the Moon is currently pulling away from Earth of course).