If a barometer was filled with milk rather than mercury, how tall would the barometer have to be to contain this solution?

Barometers are devices which are used to measure the atmospheric pressure at a given location. The basic mechanism is that a liquid rises in a tall column and the height of this liquid is converted to pressure. Mercury is the preferred liquid for such devices. For an atmospheric pressure of 1 atm, the mercury column rises to 760 mm or 76 cm. So, we can use a column of around 1 m height, to include some extra space.

If we used milk instead of mercury, the rise in the column would be different. The rise in the liquid is directly proportional to the density of the liquid. The density of mercury is about 13.5 g/ml, while that of milk is about 1.03 g/ml. Thus, a milk column will have to be about 13.1 times (= 13.5/1.03) taller as compared to the mercury column for measurement. 

Note that 1 atm of pressure will cause milk to rise by 9961 mm or almost 10 m in a column, as compared to 760 mm of mercury. 

Hope this helps.