GPS USB device with external coax antenna?
You're right 300m of USB cable is not suitable, and neither is coax from a practical point of view because even something like LMR-1700 will give you about 17dB of loss over that length. That grade of coax is normally over $10 a meter so you'd have to drop thousands on suitable cable.

You'd be better to place the receiver, antenna and a power source close to outside and transmit the data back in a digital form. RS-485 would be one robust choice for a point-to-point connection but considering how large the building is maybe you already have network infrastructure in place? In that case maybe consider a small Linux SBC that transmits data back over the existing network.