This letter studies the performance of point-to-point multiple-input-multiple-output (MIMO) systems in the presence of large but finite numbers of transmit and receive antennas. We use some recent results on the achievable rates of finite-length codes to analyze the system performance in the cases with short packets. The results are presented for different fading models. With codewords of finite length, we derive closed-form expressions for the error probability/throughput and determine the minimum number of antennas which are required to satisfy different error probability/throughput requirements. As we show, while the error-constrained required number of antennas is sensitive to the length of short codewords, its effect is negligible for codewords of moderate/long length.