Servo motors are a DC motor with a gearbox and internal closed loop feedback. They require voltage, ground, and a control signal. Based on the duty cycle of the control signal, the servo will move to that position (or speed, in the case of the continuous rotation servo).
Because they will try to move to and maintain a given position, if they encounter too much physical resistance, they can stall and demand high current (called the stall current). You need to be able to provide the high current, but also some protection in case such current is prolonged. In other words: when the motor is working against a heavy load, it will need higher current. If it gets stuck, you will want to have a means to disconnect or stop it (which may be as simple as a fuse or resettable fuse, or a controller that monitors current and cuts out as needed).
The wire you linked is 28 AWG, which is quite small. Larger currents need larger diameter wire which have less resistance and therefore lower voltage drop. If you look at a wire resistance chart such as this, you can see that 28 AWG copper is about 213 mΩ per meter. In an 8 meter length of cable you're effectively adding about 3.4 Ω of resistance. (Remember, both the voltage supply and ground form the loop, so that's 16 meters of wire.)
If all you did was leave the motor out and shorted that end, there would be 1.76 A of current flowing. (I = E/R) The motor may need more than that to perform well.
Increasing the wire gauge to 12 AWG which has a resistance of 5.2 mΩ per meter, reduces the total resistance to 83 mΩ (about a 40-times improvement). Now with a short, the available current is 72 A — more than enough.
Knowing this, you will want to do the following:
- Use thicker wire (lower AWG #) for less resistance.
- Keep power wiring as short as possible for low voltage/high current applications.
If possible, have your 6 V power supply much closer to the motor(s) and run separate wire to each motor depending on its needs. For example, if a particular motor needs 1-2 A and you limit the wire length to 1 meter, 28 AWG might be suitable.
Finally, the control signal for servos is very low current, so you don't have to increase its diameter in the same way. Obviously for long distances, you may need to use a thicker wire to impose less resistance as well.
All of this means that the typical 3-wire connectors used for servo motors are not going to accept the wire gauge you will need for long wire lengths. From the power supply to the motor will need to use bigger wire and connectors; basically whatever you want to use if you need to be able to connect/disconnect it. You might consider using commonly-available automotive or RC hobby connectors that are rated for higher current. At the motor connection, you can use the existing wires, but you'll have to decide if you need a connector there as well (connecting two dissimilar wire gauges is a little more troublesome). I would be tempted to remove the connector, strip the wire ends, and solder them together, or use some sort of splicing connector like a terminal block or "lever nut."