Despite the intensive recent research on wireless single-channel full-duplex communications, relatively little is known about the transceiver chain nonidealities of full-duplex devices. In this paper, the effect of nonlinear distortion occurring in the transmitter power amplifier (PA) and the receiver chain is analyzed, alongside with the dynamic range requirements of analog-to-digital converters (ADCs). This is done with detailed system calculations, which combine the properties of the individual electronics components to jointly model the complete transceiver chain, including self-interference cancellation. They also quantify the decrease in the dynamic range for the signal of interest caused by self-interference at the analog-to-digital interface. Using these system calculations, we provide comprehensive numerical results for typical transceiver parameters. The analytical results are also confirmed with full waveform simulations. We observe that the nonlinear distortion produced by the transmitter PA is a significant issue in a full-duplex transceiver and, when using cheaper and less linear components, also the receiver chain nonlinearities become considerable. It is also shown that, with digitally-intensive self-interference cancellation, the quantization noise of the ADCs is another significant problem.