Let's say you have n numbers and the sum of the rounding errors is X. A Uniform[-0.5, 0.5] distribution has a mean of 0 and a variance of 1/12; so the mean of X is 0 and its variance is n/12.
I think we can safely assume by the Central Limit Theorem that X has an approximately Normal distribution. So you want to find n such that
You know the mean and variance of X, so ....