For many practical purposes, e.g. aeronautical engineering applications and the dynamics of stars within a galaxy, Newton's law of gravity (i.e. G*M*m/r^2 in the case of two point masses at distance r from each other), proposed in the 17th century, remains perfectly accurate to the limits of experimental accuracy or engineering requirements.
General relativity modifies Newtonian gravity in circumstances where objects are moving with linear momentum or angular momentum that is non-negligible relative to the speed of light (e.g. special relativistic gravitational effects), in very strong fields such as those found in the vicinity of black holes, neutron stars, binary systems, and around the time of the Big Bang (singularities and other strong gravitational field effects equivalent to similar accelerations), in the vicinity of very heavy objections in circular motion (frame dragging), and in that photons have gravitational interactions (gravitational lensing).
One of the only general relativistic effect observable in an Earth bound, experimental setting (without involving astronomy observations) is the impact that the elevation of an atomic clock has on the rate at which it ticks, due to the weakening strength of the Earth's gravitational field as one's distance from the center of the Earth increases.
Where it has been possible to make experimental observations of these predicted general relativistic deviations from Newtonian gravity, they have confirmed general relativity with a small cosmological constant to the limits to experimental accuracy (which, alas is nowhere near as precise as the experimental tests of electroweak phenomena, for example), with the exception of the phenomena attributed in astronomy observations to dark matter effects, at scales from about ten meters to Earth orbit to galaxies to the scale of the universe as a whole.
In general, general relativity does not predict measurable phenomenological differences from Newtonian gravity for objects made of laboratory scale objects made of matter, at laboratory distances, at velocities relative to each other that are tiny relative to the speed of light, after controlling for any gravitational impact of the Earth's gravitational field (or any other dominant gravitational field in the vicinity of the experiment).
But, for a variety of reasons, including the weakness of the gravitational force between laboratory sized masses, it is very tricky, experimentally to measure deviations from Newtonian gravity at very short distances in laboratory experiments.
Why Look For Short Range Non-Newtonian Gravitational Effects?
Yet, as scientists, we would like to validate the correctness of the physical laws that govern gravity, one of the four fundamental forces of Nature, over as many orders of magnitude as possible. And, conceptually and theoretically, there are plausible reasons to think that at a sufficient small scale (e.g. the Planck scale of ca. 10^-34 meters, or even the atomic scale of ca. 10^-15 meters) that quantum interactions might cause gravity to behave in a non-classical manner.
Almost no realistic laboratory experiments can hope to directly measure gravitational effects between laboratory scale or smaller objects at these distance scales, other than to conclude that they do not have a measurable effect on experiments conducted at this scale relative to the Standard Model forces.
But, many beyond the Standard Model theories seek to include the three forces of the Standard Model and the force of gravity into a single unified framework, often as manifestations of some single underlying principle or particle. This isn't easy, because the three Standard Model forces are profoundly stronger than the force of gravity at a fundamental particle or atomic scale. Gravity is about 10^-28 times weaker than the weak nuclear force, about 10^-34 times weaker than the electromagnetic force, and 10^-36 times weaker than the strong nuclear force at this scale. But, there is a way to accomplish this goal, in a mathematically elegant way that is not contradicted by any current experimental evidence
First, assume that the Standard Model particles and interactions are confined to a space-time vacuum membrane with three dimensions of space of a scale not less than about 13 billion light years in extent, and a familiar dimension of time of not less than about 13 billion years in extent. Then assume, that at every point in conventional space-time, additional dimensions (frequently seven) extend for only a finite distance on the order of a small fraction of a meter (sometimes called, for example, "compactified" or Kaluza-Klein dimensions).
In this important class of beyond the Standard Model theories that include most variations on string theory and many other important classes of beyond the Standard Model theories, gravity should start to behave in a non-Newtonian manner, not at the Planck scale, but instead at the characteristic extent of the extra compact dimensions of the theory (often assumed, for sake of simplicity to all have the same size). Thus, it is not safe to assume that there will be a desert of new gravitational physics between the Planck scale and the planetary scale in the absence of experimental confirmation (although, of course, almost everyone does precisely that the vast majority of the time).
For example, if the typical extra dimension accessible only to gravitational interactions at a given point had a characteristic scale of a micrometer (i.e. 10^-6 meters) then one would expect to see non-Newtonian gravitational effects at distances smaller than a micrometer between laboratory scale or smaller objects.
Gravitational effects are the primary means by which we observe the topology of the universe at every scale.
Particle collider experiments such as the Large Hadron Collider, however, as of 2012, appear to rule out extra dimensions down to scales of about 10^-14 meters to 10^-15 meters (albeit in a mildly model dependent ways), which is about the size of a typical atom to the size of a typical molecule.
This is pretty discouraging for someone looking for theoretical motivations to look for non-Newtonian gravitational effects in experiments that can only measure gravitational effects at larger distances. Non-Newtonian gravitational effects between the classical physics scale and the atomic scale would have to be due to some other kind of theory than one predicting extra dimensions in a relatively conventional way.
This is also rather discouraging considering that extra dimensions are a characteristic feature of a huge swath of the most popular beyond the Standard Model physics theories, although the woes of these theories in the face of this evidence can often be cured mathematically with only mild tweaks to their parameters. Indeed, one can even argue in the case of many models, with a straight face, that only smaller extra dimensions as "natural" in some sense.
Still, for whatever it is worth, I'm not terribly impressed with the notion of an eleven dimensional world with four effectively infinite dimensions and seven atomic scale or smaller dimensions, at least to the extent that all of these dimensions are space-like. This seems horribly contrived to me, as does the assumption that the universe needs a full complement of superpartner particles of which we have never seen any significant experimental evidence.
Current Short Range Constraints On Non-Newtonian Gravitational Effects
To how small of a scale has the accuracy Newtonian gravity been experimentally validated, and how precise is that validation?
For example, the correctness of Newtonian gravity (which in this context is indistinguishable from general relativity) has not been experimentally validated at the very short atomic and molecular scales where chemistry and nuclear physics play out, except to the extent that we know that it is small enough to be negligible relative to the Standard Model forces.
As of early 2013, experimental evidence rules out gravitational effects more than a few orders of magnitude stronger than those predicted by Newtonian gravity at distances of less than about 10 micrometers (i.e. 10^-5 meters). The linked pre-print also explains in technical detail why it is so hard to be more precise.
By way of example, this is roughly the precision to which interchangeable motorcycle engine and firearm parts are crafted. So we know experimentally, for example, that gravity does not behave weirdly in any significantly measurable way at the scale of metal or paint powders interacting with precision crafted mechanical or electronic equipment. At these scales, only the most precisely understood force of the Standard Model (electromagnetism), and the overall, locally uniform at these distance scales, strength of the background gravitational force of the Earth-Moon system matter.
Gravitational effects dozens of orders of magnitude stronger than those predicted by Newtonian gravity are ruled out at distances of about 10^-9 meters or more, which is to say, basically, that at scales at least that large, gravity is still profoundly weaker than any of the Standard Model forces, (although this conclusion is pretty trivial conclusion in the case of the weak nuclear force and strong nuclear force that operate for all practical purposes only at extremely short, basically subatomic distances in any case).
Put another way, gravity doesn't behave weirdly at almost any distance scales large enough to be dominated by classical mechanics, as opposed to quantum mechanics, with something of a gray area at distance scales where quantum mechanical effects are tiny, but not so small that they can be completely ignored without serious thought in high precision applications.
Of course, deviations from Newtonian gravity at small scales don't have to be big (i.e. the orders of magnitude differences that current experimental constrains at micrometer and smaller scales involve) to be very interesting to scientists.
In practice, five standard deviation direct experimental proof of even a 1% deviation from Newtonian gravity/general relativity at any scale would revolutionize physics as we know it and win its discoverer a Nobel prize and fame almost on a par with Newton and Einstein. Yet, it isn't truly inconceivable, given only our experimental data, that such a subtle deviation from Newtonian gravity could take place even at the millimeter scale. So, the existing constraints on new gravitational physics at small distances are really very modest indeed.
Bonus: What Would Be The Wavelength and Amplitude of Gravitational Waves?
The range of gravitational wave frequencies that could conceivably be detected experimentally and that would have a conceivable astronomy source ranges from about 10^-7 Hertz to 10^11 Hertz (a Hertz is number of wavelengths per second, and gravitational waves would propagate at the speed of light in general relativity). A light second is about 186,000 miles (about 982,090,000 feet).
A 10^-7 Hertz gravity wave would have a wavelength of about 31% of a light year (about a tenth of the distance to the nearest star to Earth other than the Sun). A 10^11 Hertz gravity wave would have a wavelength of about 1/8th of an inch. In this direction, limitations on the minimum size of a gravity wave come from a lack of sufficiently high energy sources for it in the universe.
The amplitude of a gravitational wave in general relativity that we might expect to encounter passing through Earth might squish or stretch the fabric of space-time by a factor on the order of one part per 10^20 between its peak and trough, which relative to the radius of the Earth would be a distance of about 100 atoms in length (e.g. the length of one strand of DNA within a cell's nucleus).