What is the proof that the universal constants ($G$, $\hbar$, $\ldots$) are really constant in time and space? Cavendish measured the gravitation constant $G$, but actually he measured that constant on the Earth. What’s the proof that the value of the gravitation constant if measured on Neptune would remain the same? What’s the guarantee of its being a constant?
There are many such constants; I just took one example so that you can understand what I intended to ask. Is the value of the speed of light really a constant? Who knows that it wouldn’t change its value on other planets, or, more precisely, what’s the guarantee that the speed of light remains the same even near a black hole? If no one can ever reach a black hole, then how can a scientist claim the value of the speed of light?
 A: A recent measurement suggest that the ratio of the electron and proton masses has not changed for at least 7 billion years.
Abstract:

The standard model of physics is built on the fundamental constants of nature, but it does not provide an explanation for their values, nor requires their constancy over space and time. Here, we set a limit on a possible cosmological variation of the proton-to-electron mass ratio μ by comparing transitions in methanol observed in the early universe with those measured in the laboratory. Based on radio-astronomical observations of PKS1830-211, we deduced a constraint of ∆μ/μ = (0.0 ± 1.0) × 10−7 at redshift z = 0.89, corresponding to a look-back time of 7 billion years. This is consistent with a null result. 

A: This is more of a philosophical question. There is no way to actually prove something, see the Münchhausen Trilemma. The best we can do in science is coherence, that the theory fits the observation. Varying "constants" on other planets or near the black hole simply don't fit the data. Furthermore science is evolving and subject to change. If one day the theory no longer fits the observation, we develop a better theory.

About the units of constants:
Some would say that varying constants with units are meaningless, since you can just redefine units to fix the physical quantity. However, a certain unit can have multiple definitions not related to each other, at least in current theory. For example, unit of mass can be defined by fixing $G$, or by fixing the mass of electron $m_e$, or the mass of ${}^{12}\text{C}$. 
A: If a tree falls in a forest when no one is there to hear, does it make a sound?
If you take two measurements of a quantity and the value is the same, what was it's value between your measurements?
As described in the other answers, your question is really more philosophy than physics.  There are, nonetheless, very compelling reasons to think that the universal constants are constant.  The primary one being that theories in which they are assumed to be constant make accurate predictions about the real world and provide coherent explanations for observed phenomena.
To the first of these, the assumption that the speed of light is constant leads to a number of startling conclusions that have been largely explored and formalized in the theories of relativity.  When these theoretical frameworks are compared to observation, their predictions closely match.  This is a process of extrapolation.
To the second, when observations have been made, we can hypothesize what events led to them.  Assuming that certain aspects of the universe are constant provides a logically coherent description.  This is a process of interpolation.
By the way, if you want to see a bit about what kind of work goes into measuring these constants, you should check out the latest CODATA publication.
A: One should separate the question into two parts, the first of which is philosophical, and the second physics. The philosophical question is resolved by understanding that there are "constants" which are just those that set the system of units, and these are constant for the simple reason that they define our conventional units.
The unit-defining constants philosophically cannot change. They can only be determined relative to physical measurements using physical atoms and light, and these measurements serve to fix our units. The constants which are philosophically incapable of changing are listed below:


*

*The speed of light c, which defines the unit of space given the unit of time.

*Planck's constant, $\hbar$, which defines the unit of mass-energy in terms of the unit of inverse time.

*Newton's constant, which defines the unit of mass-energy in terms of the unit of space (and in conjunction with the other two, fixes a unique unit of mass, length, and time, the Planck units)

*Boltzmann's constant, which defines the Kelvin in terms of the Joule.

*electromagnetic constants, which define the unit of charge


In terms of Plack units, all physical constants are dimensionless. These are the quantities which are philosophically capable of changing (see this question: units and nature )
So the gravitational constant simply cannot change. It is philosophically meaningless to say that it does change. What you would really be saying is that atoms are changing size relative to Planck units.
Here are some constants that can, in principle, change:


*

*The charge of the electron in Planck charges (the square of this is called the fine structure constant).

*The mass of the proton in Planck masses (this is more or less the exponential of the strong coupling at the Planck scale)

*The Higgs VEV: this is one unnaturally small parameter in Planck units.

*The cosmological consntant: this is the other unnaturally small parameter.


The other dimensionless constants are rougly of the expected size. The electron-Higgs coupling is a bit small, so the electron is somewhat light compared to other lepton and quark masses, but to 1 part in a thousand, not one part in a billion, so it could still be a coincidence.
Within string theory, all of these dimensional constants are quantities which can change, they are all associated with a particle which represents fluctuations in these quantities. These particles are determined by the geometry of the microscopic space-time. The constants which are constant are those whose low-energy dynamics fixes their value, so that small fluctuations return to where they started, and any change in their value requires energies of order the Planck energy.
At low energies, or outside of string theory, the principle that fixes the charges and masses of the particles is renormalizability considerations. So that the reason the electron charge does not vary is that if it changes from place to place, it is a field, and no field can couple in a renormalizable way to the photon and electron-positron field. They are already dimension 4.
The principle of renormalizability tells you that the only constants you expect to see in a quantum field theory which are natural are the dimensionless coefficients of dimension 4 interactions, like the electron charge, or macroscopic scales determined by logarithmic running, like the mass of the proton. The Higgs VEV is unnatural for this reason, it is a fine-tuned mass scale, and this suggests that there is something left that we don't understand about the Higgs mechanism, which will be sorted out once we have experimental data about the Higgs boson.
The principle of renormalizability is only applicable in a scaling regime where all the energies are much lower than the Planck energy. In this regime, you also expect either Newton's constant to be truly constant, which is Einstein's gravity, or for there to be an extra massless scalar field interacting gravitationally, which is Brans Dicke theory. All other corrections are less renormalization relevant, and scale away at low energies (although Einstein gravity itself is not renormalizable, it is the leading surviving scaling term at low energy, so the renormalizability principle still works). Experimentally, we know that Brans-Dicke fields cannot be working at solar-system scales.
Because of the philosophical freedom of choosing units, Brans and Dicke chose to express their theory in terms of the gravitational constant changing from place to place. This terminology is unfortunate. They could have just as easily framed it as the speed of light changing from place to place, and had the exact same theory. It is best to have G and c both constant, and consider their field as a new scalar field that varies from place to place, with no relation to the unit-defining constants.
A: There is no proof that fundamental constants are constant. Indeed I've seen claims that string theory allows varying constants, though I've also seen comments (I think Lubos Motl blogged on this a while back) that such arguments are wrong.
There are lots and lots of publications measuring fundamental constants and review articles of such measurements. Google for "fundamental constants site:arxiv.org" to find enough articles to keep you reading for a while. At the moment there is no evidence that any of the fundamental constants have changed. There were some observations of distant galaxies suggesting that the fine structure constant may have changed in the last few billion years, but I believe this is still inconclusive. See http://arxiv.org/abs/1202.4758 for a recent paper on this.
Later: Karsus Ren makes a good point that in the above I've implied the variation is over time rather than over space. To a degree it's impossible to separate the two. Since information from "there" only gets "here" at the speed of light we can only see the constants as they were in the past. However we do observe that the universe is isotropic on the large scale, and this implies that the fundamental constants don't vary detectably on the large scale. On the solar system scale any variations in the gravitational constant would be easy to detect and none have been found so far.
A: In the Cavendish experiment we wish only to explore the motion in the horizontal direction. 
This motion should not be influenced by (little) g which is why in the experiment you need to wait (a long time) for the whole thing to settle down before you can determine much. Once it has settled down, a value for G can be determined (with a certain amount of confidence that g is not interfering).
The error associated with an experimentally determined result (like Cavendish's), tells us how precise the result is. As far as I know, the experimentally determined constants all have some uncertainty associated with them.
What you seem to be asking about though is how accurate the data is in the first place, rather than how precise it is. There is a reason they prefer to use language like hypothesis, theory and law to describe results in physics. They reason is because nothing is 100% certain. 
A: in my opinion, the concept of Planck Units sorta settles what "constants" might meaningfully vary from those that it doesn't matter because we wouldn't know the difference.
express everything in terms of Planck Units.  then $c, \hbar, G, \epsilon_0, $ and $k_B$ do not even exist to vary.  they are all $1$ except (from convention) $\epsilon_0 = \frac{1}{4 \pi}$.  (i would prefer a convention where $4 \pi G = 1$ and $\epsilon_0 = 1$).
now, the measurement or expression of every other physical constant in terms of Planck Units is a dimensionless value.  it's variation means something.  i thought John Baez was able to condense the set of fundamental physical constants (whose value could conceivably change and it would mean something) down to 26.  $G, \hbar, c$ are not on the list.  i think, with enough physics, any physical constant expressed in terms of its Planck unit, can be expressed as a function of some subset of the 26.
A: FYI, in Physics we don't really take questions like "what if..." into consideration. Instead there is magic way to prove assertions by necessarily making use of Mathematical concepts and theorems to conclude what the theory tells about. But make sure, although theories claim to be right, it doesn't mean they has to be, instead by time, someone may claim as wrong to those theories by proving with enough mathematical theorems/derivations. Same happens with constants, their values can be altered or corrected by new values brought from researches to improve for example accuracy of dependent other theories/constants on those.
