What gauge wire is needed to feed a 70 ampere panel 300 ft. away?
If you were simply concerned with ampacity of the conductors, you would be able to use 4 AWG copper or 3 AWG aluminum. Unfortunately, in your situation you'll have to concerned with the power quality, so you'll also have to take into account the voltage drop along the conductors.
Calculating voltage drop uses the simple formula Voltage Drop =  Resistance (R) * Current (I). However, figuring out resistance and current can be a bit tricky.
Resistance
To determine the resistance of a conductor, you can use Table 8 from Chapter 9 of the National Electrical Code (which you should be able to find online somewhere). This table lists the ohms per thousand feet, of various types and sizes of conductors.
Once you have this value, you can divide it by 1000 to get the ohms per foot. You'll then multiply this value by 2 times the length of the conductor. So the formula for resistance will look something like this.
R = 2 * Length (L) * ohms/foot
Current
The current is a bit trickier, since it will change depending on what is drawing power on the circuit.  A lot of folks will say  things like &quot;It' highly unlikely you'll ever use the circuit at full load.&quot;. They'll then use a fraction of the circuit capacity to calculate the voltage drop. However, this can easily lead to the situation where a high load item turns on, and the lights dim.  When calculating the voltage drop for feeders, I like to use the full capacity of the circuit to do the calculations. So in your case, I'd use 70 amperes.
Power Quality
It's recommended to only allow a 5% total voltage drop, from service to consumer. Because of this, you'll want the voltage drop along the feeder to be around 2 - 3%. For a 240 volt circuit, that means 4.8 - 7.2 volts (120 volt circuit = 2.4 - 3.6 Volts).
Examples
With all that said, let's look at some actual numbers. I'll be using values from the uncoated copper column of Table 8 for these examples.
4 AWG Uncoated Copper
VD = 2 * 300 * 0.000308 * 70 = 12.936 Volts
12.936 Volts / 240 Volts = 5.4%
3 AWG Uncoated Copper
VD = 2 * 300 * 0.000245 * 70 = 10.29 Volts
10.29 Volts / 240 Volts = 4.3%
2 AWG Uncoated Copper
VD = 2 * 300 * 0.000194 * 70 = 8.148 Volts
8.148 Volts / 240 Volts = 3.4%
1 AWG Uncoated Copper
VD = 2 * 300 * 0.000154 * 70 = 6.468 Volts
6.468 Volts / 240 Volts = 2.7%
1/0 Uncoated Copper
VD = 2 * 300 * 0.000122 * 70 = 5.124 Volts
5.124 Volts / 240 Volts = 2.1%
Depending on how much you actually care about power quality, you'll want to use either 2 AWG copper or larger. I'd personally probably use at least 1 AWG, because I know how power demand tends to grow over time. Once you have power in a location, you'll find all sorts of things to plug in.