We present an efficient method of utilizing pretrained language models, where we learn selective binary masks for pretrained weights in lieu of modifying them through finetuning. Extensive evaluations of masking BERT, RoBERTa, and DistilBERT on eleven diverse NLP tasks show that our masking scheme yields performance comparable to finetuning, yet has a much smaller memory footprint when several tasks need to be inferred. Intrinsic evaluations show that representations computed by our binary masked language models encode information necessary for solving downstream tasks. Analyzing the loss landscape, we show that masking and finetuning produce models that reside in minima that can be connected by a line segment with nearly constant test accuracy. This confirms that masking can be utilized as an efficient alternative to finetuning.