how do you calculate the cube root without calculator

The iterative method Bertie mentioned, to find the cube root of "a", say, is:

x_{n+1}=\frac{1}{3}(\frac{a}{x_n^2}+2x_n)$$

This is derived using Newton-Raphson.

However, in general, I wouldn't want to use this without a calculator of some sort! 

There is, I vaguely remember, an algebraic method similar to that for calculating square roots and it's also possible to use the Newton-Raphson method, (though you would need to be good at arithmetic, hardly the sort of calculation you would attempt without a calculator).

If the number is close to some convenient cube, it's possible to make use of the binomial expansion. For example.

$$\begin{array}{lcl} \sqrt[3]{1005}&=&(1000+5)^{1/3}\\ &=&1000^{1/3}(1+0.005)^{1/3}\\ &=&10(1+(1/3)0.005+(1/3)(-2/3)0.005^{2}/2!+\dots)\\ &=&10(1+0.0016667-0.0000028+\dots)\\ &\approx&10.016639 \end{array}$$

That's easily done on paper and is correct to 6dp.

That's an interesting tecnique, Alan and Bertie......

I'm with Alan...........a calculator seems vital......