This is a standard question in diophantine approximation.  See for example Chapter 3 of Waldschmidt's book <i>Diophantine Approximation of Linear Algebraic Groups</i>.  Here is the simplest bound that Waldschmidt gives.
Assuming $a_0\ne 0$, let $H := \max_{0\le i\le n} |a_i|$.  We will show that if $\alpha$ is any nonzero root of your polynomial, then $|\alpha| > 1/(H+1)$.

If $|\alpha| \ge 1$ then the inequality trivially holds.  Otherwise, if $|\alpha|<1$, then

$${1\over |\alpha|} \le \left|{a_0\over \alpha}\right| = \left| a_1 + a_2\alpha + \cdots + a_n \alpha^{n-1} \right| \le H(1+|\alpha| + \cdots+|\alpha|^{n-1}) < {H\over 1-|\alpha|},$$
which rearranges to $|\alpha| > 1/(H+1)$.