Let $$Z_1, \ldots, Z_n\sim \begin{cases}+1&\text{w.p. 1/2}\\-1& \text{otherwise}\end{cases}$$ be independent Bernoulli variables and let $S_n=Z_1+\ldots+Z_n$ be the result of an unbiased $n$-steps random walk.
Clearly, $\mathbb E(S_n)=0$ and $Var(S_n)=Var(Z_1)+\ldots+Var(Z_n)=n$.
Thus, we can use Chebyshev's inequality and get $$\Pr\left[S_n\ge k\sqrt n\right] \le k^{-2}.$$
However, I'm interested in a lower bound on the probability that $S_n\ge k\sqrt n$ for a fixed $k=O(1)$. Specifically,
Can we find a function $f(k)$ such that for all $n$, $\Pr\left[S_n\ge k\sqrt n\right] \ge f(k)$?