Square roots of non-squares are irrational.
Square roots of numbers between those two numbers will be between them,
So let's find some non-squares between their squares, and get their square
roots:

Their squares are:

 = 0.5102040816

 = 0.6694214876

Some non-squares between those are 

2/3 = 0.66666667
3/5 = 0.6
4/7 = 0.571428
5/8 = 0.625
5/9 = 0.55555

Their square roots will be irrational and between  and 

 = · = 
 = · =  = 
 = · =  
 =  =  

Edwin