Let $\mathcal{C}_{f_0}$ and $\mathcal{C}_{f}$ denote the spaces of almost null and almost convergent double sequences, respectively. We show that $\mathcal{C}_{f_0}$ and $\mathcal{C}_{f}$ are BDK-spaces, barreled and bornological, but they are not monotone and so not solid. Additionally, we establish that both of the spaces $\mathcal{C}_{f_0}$ and $\mathcal{C}_{f}$ include the space $\mathcal{BS}$ of bounded double series.