Beck-Katz does:

Var(b) = A^{-1} W A^{-1}

where A = \sum_{i=1}^n X'_i X_i,
W = \sum_{i=1}^n \sum_{j=1}^n \sigma_{ij} X'_i X_j
and \sigma_{ij} is estimated as (1/T) \sum_{t=1}^T e_i e_j,
with the e's being OLS (or fixed effects) residuals.

The trouble is there's no guarantee that W (which is supposed
to be a variance measure) is positive definite in unbalanced
panels; if it's not, we "fail".

> Also, which standard errors are reported instead?

The "classical" ones. If the Arellano method is used you'll
see "Robust (HAC) standard errors".