I'm trying to solve a problem that goes like this:
I've made measurements of a physical quantity that is normally distributed. I estimate the mean and standard deviation, with some uncertainty. How many measurements do I need to make to determine the standard deviation with an accuracy of 30%, 10% and 3%?
I've searched for formulas I can use in the class materials and the books the teacher recommends but to be honest I'm not even sure of what I'm looking for.
From the iid sample $X_1,...,X_n\sim N(\mu,\sigma^2)$ define the estimator of $\sigma^2$ as $$ \hat{\sigma}^2_n=\dfrac{1}{n}\sum_{i=1}^n(X_i-\overline{X}_n)^2. $$ I interpret from your problem that you want $n$ such that
$$ P\left(\Big|\dfrac{\hat{\sigma}_n-\sigma}{\sigma}\Big|\leq \epsilon\right)\geq 1-\alpha $$ for some precision $\epsilon> 1$ and some "accuracy" $\alpha\in(0,1)$.
It is a known fact that (for gaussian samples) $$ n\dfrac{\hat{\sigma}^2_n}{\sigma^2}\sim\chi^2_{(n-1)}. $$
Choose the sample size $$ n = \dfrac{q_{1-\alpha}}{(1-\epsilon)^2}, $$ where $q_{1-\alpha}$ is such that $P(\chi^2_{(n-1)}\leq q_{1-\alpha})=1-\alpha$. Therefore, \begin{align*} P\left(\Big|\dfrac{\hat{\sigma}_n-\sigma}{\sigma}\Big|\leq \epsilon\right)&\geq P\left(\dfrac{\hat{\sigma}_n}{\sigma}+1\leq \epsilon\right)\\ &=P\left(n\dfrac{\hat{\sigma}_n^2}{\sigma^2}\leq n(\epsilon-1)^2\right)\\ &=P\left(\chi^2_{(n-1)}\leq q_{1-\alpha}\right)\\ &=1-\alpha. \end{align*}