About driving LEDs over long distances
There are two very different wire pairs which you need to consider.

The first is the power pair, which provides 5 volts DC. This is your main problem, and the general answer to your question is that you can drive 5 volts over pretty much any length you choose, as long as you make your wire thick enough. You can do this by

a) Calculate the effective resistance of your strip. Divide the voltage (5 volts) by the current (in amps) to get a resistance in ohms. 

b) Determine the length of cable, and multiply by 2. 

c) Look up a table of wire gauges, and pick one which will have a total resistance over the wire length less than 10% of the resistance you found in a). And going for an even smaller resistance won't hurt, either.

d) Finally, remember to put a minimum 100 uF capacitor across the 5 volts at the strip end of your cable.

Now for the other pair - the signal pair. Since this has a load of (nominally) about 120 ohms, it can be much thinner than the first pair. You can use the procedure above to check your wiring size, except that you must leave out the cap.

However, what's important is not so much the wire thickness as the signal quality. Twisted pair is a minimum, and shielded twisted pair even better. I don't know what bit rate you expect to drive the strips with, but if you pay attention to what you're doing I'd estimate at least 100 meters is possible.

ETA - If you look at the 75176 data sheet, fig 10.2.3, you'll see that 5 MHz at 100 ft will give usable results. Since you're driving with an Arduino I'd guess that you won't be providing anything like 5 MHz, so your cable can be longer than 100 feet, and my number of 100 meters still seems reasonable. Of course, this requires that you use a very good signal cable such as the CAT 5 specified in the data sheet. If you are, your best bet might be to use one pair in a cable as signal, and parallel the other 3 to provide 5 volts.