I measured something $N$ times using different measurement techniques. Each measurement technique $i$ has a known variance $\sigma_i^2$. So every measurement is $x_i = \hat x + \epsilon_i$ where $\hat x$ is the true value and $\epsilon_i$ is pulled from a normal distribution with mean 0 and variance $\sigma^2_i$.
Ok, I have all these measurements. Now I want to know the variance of my measurements taken together, taking into account the fact that I know the variance of all the measurement techniques. What's the best way to do this?
You can just treat them as sum of normally distributed variables, which is extensively written down in Wikipedia.
In short, if $X \sim N(x,\sigma_x^2)$ and $Y \sim N(y,\sigma_y^2)$ and $X,Y$ are independent, then $$X+Y \sim N(x+y, \sigma_x^2 + \sigma_y^2)$$
In your case you have the measurements $X_i \sim N(x, \sigma_i^2)$, but you don't know the actual value $x$ (which is why you're doing the measurements.) Now In order to find $x$, you probably want to take the average of all of them, that is $\hat x = \frac{1}{n} \sum_{i=1}^n X_i$
If we now treat $\hat x$ as a random variable, we can determine its variance by using the formula above, and we get:
$$n \hat x = \sum_i X_i \sim N(nx, \sum_i \sigma_x^2)$$
Now as a last step you have to "solve" for $\hat x$ which is not difficult. (Use that $Var(aX) = a^2 Var(X)$, and $E[aX] = aE[x]$)