Yes!! Get safety glasses & sledge hammer, and gear up for a work out. In my opinion you'll need to address more than just the shelves, the mantel & hearth are integrated with the shelvse. Rip down shelves and remove mantel & rip bout hearth to fireplace surround. Once you have all of this cleared away you'll have a clearer idea of what is possible. It may be as simple as a simple wall unit or a more complicated built in cabinet.. Send along pix after you take down the shelves& rip back the hearth! Good luck!

Perhaps attach hinges and neat doors without handles and turn them into hidden storage?

No, I don't think you need to rip it out. You'll be adding something back! I agree witj j22b to style it attractively and if you can add lighting it would shine nicely!

Don't rip it out! As j222b says style it more attractively and some lighting.

I disagree with J222b & Darzy, unfortunately. While the feature is very unique and styling & lighting would make it more attractive, the bulkiness of these shelves are going to make it difficult to bring in any modern or bright feeling you might be going for. I do recommend demolishing the shelving, but I'm kind of fond of the chunky mantle. I think that replacing the shelving with something a little slimmer in a configuration that makes sense with the function of the space is your best bet. But still add lighting!! Good luck!