Can many independent weak predictors (with low corr. with the target) act together and result in high accuracy in a linear model? Is it even possible? The question just came to my mind. I notice that generally succesfull models include at least some predictors that somewhat correlate to the target. But is it completelly necessary? Is it possible to succesfully predict the target using many predictors with very low correlation (e.g. corr < 0.1)
 A: Yes, this is possible
This is actually a common phenomenon in regression analysis.  Broadly speaking, if predictors are not too correlated amongst themselves (so that they are not duplicating information too much), they will tend to add information to the regression even if they are individually adding only a small amount of information.  With a sufficient number of (individually) weak predictors you can make a prediction of the response variable that is highly accurate.  The overall accuracy is somewhat complicated --- in linear regression it depends on the full correlation matrix of the predictors and their individual correlations with the response variable (see O'Neill 2019 for a geometric analysis).
As an extreme example of this phenomenon, it is possible to have a set of explanatory variables that are weak predictors individually, but they are collectively perfect predictors of the response variable.  For example, suppose we take some large number of predictors $m$ and we set:
$$x_1,...,x_m \sim \text{IID N}(0, \sigma^2)
\quad \quad \quad \quad \quad y \equiv \sum_{k=1}^m x_i.$$
In this case, the vector $\mathbf{x} = (x_1,...,x_m)$ perfectly predicts the value $y$ even though individually we have correlation values:
$$\mathbb{Corr}(y, x_k) = \frac{1}{\sqrt{m}}.$$
If we take $m$ to be large then these individual correlation values for the predictors are small, even though the vector of predictors perfectly determines the response.  (Taking $m > 100$ is enough to get $\mathbb{Corr}(y, x_k) < 0.1$ as desired in your question.)
