This "science is simply wrong" idea is also a popular way to deal with the implications of galactic and intergalactic distances, as well as many other ancient artifacts in natural history. Science can always be wrong. However, this must be determined on a case by case basis based upon the available evidence and laws of nature.

There are a couple of ideas one might have in mind.
1) Starlight travelled much faster in the past than now so that such great distances could be covered in much shorter times.
2) The distance to the LMC is simply wrong. It is only a few light years away.

The problem with the first idea is that the speed of light is not variable but a constant: 299,792,458 m/s. ⁽¹⁾
Some have suggested that the speed of light is slowing down in a roughly exponential fashion (B. Setterfield).⁽²⁾
There are two major problems with that suggestion: it violates mathematics and is contradicted by data.⁽³⁾

[The reader can skip this mathematical paragraph if desired.]
The speed of light is not just a constant based upon empirical measurements, but is a mathematical necessity.
James Clerk Maxwell worked out the four equations for light in the late 19th century.⁽⁴⁾ These are a class of mathematical expressions known as vector differential equations. There is a known mathematical method for solving such differential equations. The method yields an answer in the form of a set of algebraic expressions about the electric and magnetic field strength, plus four mathematical constants. The value of the constants must be determined by measurement, but the fact that they are constant comes from the math itself. These equations turn out to be descriptions of wave motions. Hence, we consider light in most applications simply as a wave of energy. (Quantum effects such as for a CCD video camera require some additional equations.) We can easily see that waves move with some speed simply by watching waves at the beach. Upon closer observation one can see that the speed of waves is constant (given the same type of water "waving"). The same is true for electromagnetic waves. As it turns out one of the constants that falls out of Maxwell's wave equations represents simply the speed of the wave in empty space -- it has the units of distance per time, the definition of speed. One of the variables that comes in the solution is the actual speed of the wave when travelling through some material: light slows down in materials. Therefore this speed constant is the maximum speed of light in a vacuum. It is a mathematical constant if Maxwell's equations are correct. They have never been shown to be anything but precisely correct in describing everything from radio waves to light to nuclear gamma radiation. (There are some physicists who challenge the completeness of Maxwell's equations in some situations, but not their correctness for classical applications such as star light propagation.)

Einstein's equation for speed of light is the famous E = m c^2 .  This means that energy is proportional to the speed of light squared. If the speed of light 10000 years ago was so fast to make it seem like 10 billion years ago at the current speed of light, that would mean the speed of light has changed by a factor of a million. But an increase in the speed of light by a factor of a million would mean energy increased by a million million times. Consider what a newly created earth 10000 years ago would have felt like if the sunlight hitting the earth were a million million times more energetic. Clearly, such is not possible. Geological indicators ( such as ice accumulation rate) show that the sun's energy on the earth 10000 years ago was about what it is today. Therefore, the evidence is that the speed of light was about the same 10000 years ago as it is today.

To determine the maximum speed of light in a vacuum requires measurement, not theory.  Consequently scientists have been measuring the speed of light for the past two hundred years. It is now known to such incredible precision, 299,792,458 m/s, that we no longer worry about it. If the speed of light were not a constant, if it was changing even the least little bit, we could measure the change in the speed of light over even a few months given that this parameter of light is measurable with such great accuracy. It can be measured to an accuracy better than 1 part per billion. Therefore if the speed of light had changed as Setterfield claims so that ten billion years now looks like only ten thousand years that would only be a change of 1 part per million, which is a thousand times greater than the minimum change current measurements could detect. In other words, if any changing was there to be seen, it would have been seen even by crude student-grade experiments.

Furthermore, the spectral content of starlight (its color emission and adsorption lines) allow us to measure a property called the fine structure constant (2pi e²/hc ~ 1/137).⁽⁵⁾ This depends on three constants: the quantum unit -- Plank's constant; the charge quantum -- constant charge of an electron; and, the constant speed of light, c. Since all of these are constants of nature, the fine structure constant is always the same, even in the light from the most distant stars observed. It is the same value in starlight as is measured in earth laboratory experiments. Therefore this is confirmation that all of these values are constant everywhere. This shows that the speed of light has never changed anywhere in the universe, and more importantly that the basic properties of physics (quantum electrodynamics) are the same everywhere in the universe. [Yes, it is theoretically possible that all three constants vary together so that their ratio stays the same, but that would be what we call an ad hoc argument with no basis other than it gives the answer desired. It is very improbable from a physical perspective, to put it mildly. See above comments on the relation of speed of light to energy to see why such is actually an impossibility.]

The problem with the second idea (the stars are really much closer) is that if this supernova were only a short distance away, we would be dead. If we take the age as only 10,000 yr with this event occurring about 10 yr ago, this would mean that the supernova must have occurred no more than 10 ly away from earth: 17,000 times closer than it appears to be -- and 300 billion times brighter! Ten light years from a supernova is a death zone. For comparison, the visible outer rings in the SN1987a photo are 1300 light years in diameter from precursor ejections of dust prior to the more recent visible events.⁽⁶⁾ At only 10 years away the shock wave from the explosion would have been most unpleasant here on earth. In any event, if it were much closer, say only 10,000 light years, it would be about 300 times brighter than it is: more like the Crab Nebula which is much closer (~6000 ly) within our own Milky Way and hence much brighter. (This comes from the inverse square law for isotropic radiation sources.)

The distance to the Tarantella Nebula and the Large Magellanic Cloud is well known through a technique using Cephid Variables. This technique is described by Roberts in the online book available on the AgeEvidences page that gave you this "brain teaser."⁽⁷⁾  The following links will provide more detailed information on measuring the distance to gallaxies.
        http://oposite.stsci.edu/pubinfo/pr/1999/19/index.html
        http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/970415c.html

References:
(1) CRC Handbook of Chemistry and Physics, 74th Ed., pp1-1 to 1-6.
(2) Setterfield, see ICR materials, e.g.,   http://www.icr.org/pubs/imp/imp-sub.htm , Impact #179.
(3) Fischer, analysis by a nuclear physicist of Barry Setterfield's claim that light is slowing down (extracted from a larger briefing by Fischer on ICR YEC views)
(4) Jackson, Classical Electrodynamics, 2nd Ed., John Wiley & Sons, 1972, pp217ff.
(5) Weidner & Sells, Elementary Modern Physics, Allen-Bacon, 1968, pp239.
(6) David Mahlin, Anglo-Australian Observatory,  http://www.aao.gov.au/local/www/dfm/aat048b.html
(7) Roberts, Evidences That Have Led Many Scientists to Accept an Ancient Age for the Creation of the Earth and Universe,  http://www.lordibelieve.org/page15.html , 1996.