Flexible long wire for high currents
Power companies get around it by increasing the voltage and decreasing the current.  There's nothing you can do to reduce the total gauge of wire needed to transfer a given current, but if you can reduce that current then you can reduce the total gauge needed.

Instead of feeding the 5V direct from the ATX supply you could consider feeding 12V from the ATX supply instead - you could then halve the current requirement for transmission.  On your remote device you can then use smaller light weight switching regulators (obviously rated for the right currents) to convert it to a higher current 5V supply.

Even better if you could ditch the ATX supply and use a slightly higher voltage, say 24V or 48V, which you then regulate down with switching regulators in your remote device.