Beta is not correlation

Idea. The regression splits the stock's variance into a part explained by the market and a residual part. The explained share is $R^2={\rho }^2$, so the residual keeps the share $1-{\rho }^2$.

Step 1: the beta. $$\beta =\rho \frac{{\sigma }_s}{{\sigma }_m}=0.5\cdot \frac{2}{1}=1.$$ A correlation of only $0.5$ still gives a beta of 1, because the stock is twice as volatile as the market.

Step 2: the split. The residual is uncorrelated with the market, so $${\sigma }_s^2={\beta }^2{\sigma }_m^2+{\sigma }_{\epsilon }^2.$$ In units of percent squared, $4=1+{\sigma }_{\epsilon }^2$, so ${\sigma }_{\epsilon }^2=3$.

Step 3. $${\sigma }_{\epsilon }={\sigma }_s\sqrt{1-{\rho }^2}=2\sqrt{0.75}=\sqrt{3}\approx 1.73\%.$$

Two traps. Taking the beta to be the correlation ($0.5$). And forgetting that the regression in the other direction is different: regressing the market on the stock gives slope $\rho {\sigma }_m/{\sigma }_s=0.25$, so after a $4\%$ move in the stock the best guess for the market is $1\%$, not $4\%$.