Wavelength = 300/frequency (in meters) 300 MHz center of the milair band (close enough)
So, 1/4 wave whip is 1/4 meter long. Allowing for a .95 lengthening factor due to thickness of the antenna in wavelengths, this produces a 9.3" whip.
If you honestly wanted to create a 'broadband' version, you'd build it of very high diameter tubing, say 6" dia. copper. You'd need to look up the actual formulas for adjusting for element thickness, but this would widen the antenna out quite a bit. In practical terms, for receiving, it really doesn't matter that much. And it'd be a hum-dinger of a WHAP if you hit something with that mounted 6" dia tube, too!