Listening environments contain background sounds that mask speech and lead to communication challenges. Sensitivity to slow acoustic fluctuations in speech can help segregate speech from background noise. Semantic context can also facilitate speech perception in noise, for example, by enabling prediction of upcoming words. However, not much is known about how different degrees of background masking affect the neural processing of acoustic and semantic features during naturalistic speech listening. In the current electroencephalography (EEG) study, participants listened to engaging, spoken stories masked at different levels of multi-talker babble to investigate how neural activity in response to acoustic and semantic features changes with acoustic challenges, and how such effects relate to speech intelligibility. The pattern of neural response amplitudes associated with both acoustic and semantic speech features across masking levels was U-shaped, such that amplitudes were largest for moderate masking levels. This U-shape may be due to increased attentional focus when speech comprehension is challenging, but manageable. The latency of the neural responses increased linearly with increasing background masking, and neural latency change associated with acoustic processing most closely mirrored the changes in speech intelligibility. Finally, tracking responses related to semantic dissimilarity remained robust until severe speech masking (-3 dB SNR). The current study reveals that neural responses to acoustic features are highly sensitive to background masking and decreasing speech intelligibility, whereas neural responses to semantic features are relatively robust, suggesting that individuals track the meaning of the story well even in moderate background sound.