Is a high resolution QR Code with only 5 kilobytes much too small to print on a big billboard?

No. The beauty of vector graphic is that you can scale a QR Code as big as you want without blurring the QR Code, because only polygons are saved and not pixel by pixel. Therefore only very little memory is needed and the file size is only a few kilobytes. You can download QR Codes in different vector formats like pdf, eps or svg.