What would it take for the omitted-variable bias from multiple omitted variables to cancel out? Let's stick to ordinary least squares linear regression for now, and assume the typical conditions for the Gauss-Markov theorem. If it is helpful to assume Gaussian errors, that's fine.
In such a setting, even if the OLS estimator is unbiased when the needed variables are included in the model, when a variable is omitted that should have been included, such an omission can bias the estimation of the coefficients of the included variables. This bias means estimates are too high or too low (on average).
If there is second omitted variable, there could be a second source of omitted-variable bias. If that bias is in the opposite direction as the bias coming from omitting the other variable, then the two biases could cancel out.
What would be the conditions for the biases arising from omitting multiple relevant variables to cancel out and give unbiased estimation of the included coefficients? If it helps, assume just one coefficient of true interest where unbiased estimation is desirable.

Expanding on the comment as an answer. First write the true model as
$$ Y_i = X_{1i}^T\beta_1 + X_{2i}^T\beta_2 + \epsilon_i \tag{1}$$
where $X_{1i}$ is a $K_1 \times 1$ vector of covariates (resp. for $X_{i2}$) then the reduced model is
$$ Y_i =  X_{i1}^T\gamma + \nu_i $$
Note that the OLS estimator assumes (and essentially forces via orthogonal projection in the sample) that $\mathbb{E}[X_{i1}\nu_i] = 0 $ even though this is not true by our true model. Thus the OLS estimator is giving us the sample version (assuming full rank of the entire covariate vector) of
$$ \gamma = \mathbb{E}[X_{i1}X_{i1}^T]^{-1}\mathbb{E}[X_{i1}Y_i].$$
Substitute in the true model (1) for $Y_i$ above to yield that
$$ \gamma = \beta_1 + \mathbb{E}[X_{i1}X_{i1}^T]^{-1}\mathbb{E}[X_{i1}X_{i2}^T]\beta_2. $$
Thus bias is zero when $\mathbb{E}[X_{i1}X_{i2}^T]\beta_2 = \mathbf{0}_{K_1 \times 1}$, or like you say, $\beta_2$ is in the kernel of that (sort of) covariance matrix of $X_1$ and $X_2$.
