Computability of sequential cubic-order algorithms
Estimate the constant of the leading term in your complexity, i.e., write your complexity as $c n^3 + o(n^3)$. You can get an idea of $c$ by a careful inspection of your algorithm and you can test it experimentally.

Then decide what "can be solved using today's technology" means. You probably want your program to terminate withing a certain amount of time $T$. Estimate how many operations $N_T$ your machine can perform within time $T$.

Finally solve $c n^3\le N_T$, i.e., $n \le \sqrt[3]{n/c}$.

Keep in mind that this is just an estimate there are a lot of thing that can go wrong: $c$ depends on a lot of practical factors (the language of your implementation, the target architecture, cache locality, ...), the complexity of the algorithm might not be uniform across different instances of the same size, the complexity of the algorithm might not scale up nicely, the asymptotic notation is... asymptotic while you're dealing with small input sizes, and more.
You know the details of your algorithm so you probably have a better idea of how these factors will impact your estimate.