jmc

Note that $7^3<500<8^3,$ so any positive integer that can be written as the sum of two positive perfect cubes must be written as the sum of two cubes $a^3+b^3$ where $1\le a\le 7$ and $1\le b\le 7.$ We can make a chart of the sum of two such cubes: $$\begin{array}{cccccccc}& 1^3& 2^3& 3^3& 4^3& 5^3& 6^3& 7^3\\ 1^3& 2& 9& 28& 65& 126& 217& 344\\ 2^3& & 16& 35& 72& 133& 224& 351\\ 3^3& & & 54& 91& 152& 243& 370\\ 4^3& & & & 128& 189& 280& 407\\ 5^3& & & & & 250& 341& 468\\ 6^3& & & & & & 432& 559\\ 7^3& & & & & & & 686\end{array}$$ As we can see from the chart, there are $\boxed{26}$ such numbers less than $500.$