Contact paper might work.

You could cut hardboard to size and put that in. It paints well and is fairly inexpensive. https://www.lowes.com/pd/47-75-in-x-7-98-ft-Smooth-Brown-Hardboard-Wall-Panel/3014304

Thanks for your suggestions! For the board additions, how would you suggest attaching it to the cabinets? Also, I don't have any cutting tools, what kind of equipment would i need for cutting these boards? Apologies for the basic questions.

Paint in and of itself will make an enormous difference. I would paint the backs and tops the same colour that you do the shelves and sides and then evaluate whether you need to add any kind of beadboard etc. (I'm of the "do the least amount of work as necessary" mindset for this kind of thing.) You won't recognize these when you're done, truly. And once your dishes are in you will barely see the backs.

Contact the cabinet maker. He did it the cheapest way possible. He needs to bear the brunt of fixing the issue.

Seeing your last comment, if you don't have the ability to DIY the boards you can get paper shelf liner on rolls like wrapping paper in a variety of colours. Cut it an inch or so bigger than needed all around, fold the edges under for a finished look so you can adjust it to any unevenness in the space, then paste it in place.

Thanks everyone! If we do end up painting alone, with no additional boards, do you have suggestions for what kind of paint we should use (which will work well on the exposed wall and the shelves)?

Contact the cabinet maker.

The cabinets could be 50 years old. Good luck tracking him down and forcing him to build them properly. Maybe check the obituaries first to see if it's even worth trying to contact him. :)

Our old builder cabinets had no backs either. However, they were 'painted' with one coat of spray primer and a very thin coat of flat paint. The combo was good for collecting dust and smudges and not much else so we repainted not long after we moved in. Just used eggshell/satin paint in Behr Swiss Coffee and a paint brush since it was difficult to get the roller in there. The cabinets were white inside and out so that worked. You could do similar to yours.

Yes, any very washable paint will do. Even semi-gloss would work there and bounce a little more light around in the cupboard.

Thanks again! Another follow-up question about painting, would it look strange to have the inside painted white but the doors are still wooden oak doors? I was considering either gel staining the doors or painting them, but maybe the staining would clash with the white interior? Perhaps I should stick with painting the doors?

No, the insides can be white while the outsides are wood. Remember anything is an improvement! But if you want you can do a fun colour inside instead, like blue, green, pink, yellow.

If you decide to restain, research well (you might get some tips here too) and stain first to avoid staining the paint :) Getting a good painted finish on cabinet fronts is a lot of work, but doable.

Thanks very much Lindsey! I appreciate your advice.

How old is the house and how old are the cabinets?

The house and the cabinets are from the 80s.

Paint will make a huge difference. If oil-based paint is available in your area, use it in an enamel finish. It will be hard and very easy to keep clean. Don't worry about the inside being painted - it will be fine.