Hey @MaximusDecimusMeridi, the term “feature extraction” usually means to extract or “pool” the last hidden states from a pretrained model. So fine-tuning a model for feature extraction is equivalent to fine-tuning the language model, e.g. via masked or autoregressive language modelling. (You can find a BERT-like example of fine-tuning here 6, and indeed one does not need any labelled data for this step).
For BART, the situation is a bit more complex because it is a seq2seq architecture, so you would likely need to frame your fine-tuning task in that manner (e.g. as a translation or summarization task).
Most applications that need feature extraction (e.g. neural search) perform best with encoder-based models like BERT and friends - I recommend checking out sentence-transformers  (link) which provides many state-of-the-art models for these applications