You can do simpler than using Chebyshev’s inequality : the distribution of $\sum_{i=1}^n X_i^2$ under $H_0$ known, it is a $\chi^2(n)$. From this you can get a threshold $t_\alpha$, using simply a chi square table: $t_\alpha = x_{n,1-\alpha}$, the quantile of level $1-\alpha$ of $\chi^2(n)$. 

This is a particularly simple situation as the variance of the law under $H_0$ in known. This is almost like the kind of artificial stuff I teach to beginners, I would be pleased if you were kind enough to describe the concrete situation that lead you to this model.

You can compare the threshold you get from Chebyshev inequality: for $\sum X_i^2$ this $n\left(1+\sqrt{{2n\over\alpha}}\right)$ to the $t_\alpha$ obtained by quantiles of $\chi^2(n)$, you will see that the quantiles are much lower; this will be much more powerful than you thought.