Would the landlord paint them ?

I would If painting is the solution. Or at least ask if we are allowed to.

I think you'll be sorry if you paint them. All kinds of food items can attack paint. I predict that it will look shabby in a year or two. Besides, if adhesive won't stick to the wood, probably paint won't either.

I like Izzy's vinyl tile idea. Similarly, I've used leftover sheet vinyl from floor installations for shelf liner. Don't bother with adhesive -- just cut to shape and lay it in loose. Vinyl flooring lasts forever, resists almost everything that might leak or spill on it, and wipes clean with a sponge.

Depending on how it was rolled, you may have to reverse-roll it and/or weigh down the edges for a while to get it to lie flat. One trick is to weigh it flat, warm it with a hair dryer or maybe even just in a sunbeam, and then allow it to cool flat.

If you don't have any vinyl leftovers or scraps, you may be able to get some from a flooring store for a small cost.

FWIW, I've cleaned old shelves in my rentals with a a 10:1 water:bleach mixture, or maybe a little stronger. That also sanitizes them. Obviously they need to dry thoroughly before you put anything in the cabinets.

You might also try diluted ammonia or full strength vinegar. Again, dry thoroughly.

Cover with heavy duty liner and use thumbtacks to hold in place???

Use plain white shelf paper with no sticky back. Use thumbtacks or blue painter tape to keep it in place. The blue tape can be removed quickly without damage when you move out.

Plain white shelf paper comes in rolls. It is much cheaper than Contact paper.

Like this: https://www.amazon.com/Royal-Consumer-Products-21055-Shelf/dp/B000BQ7WEK

If your budget will cover it, try the vinyl grip type shelf liner. https://www.amazon.com/Duck-Select-Drawer-12-Inch-20-Feet/dp/B002AS9NAS

It's perfectly okay to use one type of shelf paper in hard wearing areas and another type in relatively unused areas such as top shelves.

Most likely the shelves (cabinets also) are made of cheap pressed and glued saw dust panels with a very thin layer of plastic laminated over it. The stuff will not come clean and nothing will stick to it with any luck once it gets a few years old. Water and a lot of chemicals will cause the glue to break down and make the problem worse (especially if the plastic has worn through).

The best solution I have found for fixing them is to use a 5mm thick Poplar Plywood panel call RevolutionPly. It comes in 4' x 8' sheets at places like Lowes and Home Depot ($15.00) a sheet. Our local hardware store has it for a few dollars less than the big box stores.

RevolutionPly is a very smooth plywood that has two sides. One has a grain like tiger stripe and the other like a good veneer with knots. It takes paint and stains very good and is easy to cut with the correct saw blades. One 4 x 8 sheet will go a long way if the cuts are measured and laid out ahead of cutting. Once cut it can be brad nailed down using a few 5/8" brad nails around the outside edges.

There are a lot of good ideas already, and also at this old thread: https://www.houzz.com/discussions/need-advice-on-best-kitchen-shelf-liner-dsvw-vd~2459824 You might try the rubbery, non-adhesive rolls that are sold in dollar and hardware stores, although I don't like their anti-slip, cushiony properties for all cabinets. Even if you do one or two shelves a night after your two-year old is in bed, you'll be accomplishing something!

I buy pre-pasted vinyl wallpaper from garage sales & resale shops to use for shelf and drawer liners. It's cheap, easy to work with, easy to wipe clean.

Good old-fashioned oil cloth and thumb tacks still work.

BBB has some various shelf liners....

https://www.bedbathandbeyond.com/store/s/shelf-liners?ta=typeahead

We have our kitchen shelves lined with vinyl flooring....just cut to fit and you can use thumb tacks in the corners if needed. In our bath we have oil cloth with thumb tacks to keep it smooth. Easy to wipe down.

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Many years ago, I covered our shelves with adhesive backed vinyl floor tiles from Home Depot. I bought the cheapest they had and used scissors or a box cutter and ruler to trim them. They still look great and are so easy to install and clean.

Perplexed why almost everyone is suggesting a band-aid fix to the problem rather than a solution. Why not try to clean them first? Then you can do any of the above suggestions if you wish.

Cleaning them will take less time than cutting and fitting all the drawers. Depending on what exactly is in them (food?), covering them before cleaning is asking for problems. Mold comes to mind.

I have yet to encounter something that can’t be taken care of with vinegar + baking soda.

@lindsey_9002 great name! I just assumed the OP tried to clean...

OP, I bought a roll of paper shelf liner at a big box store. Similar to gift wrap but sturdier and wider. You can cut it to a bit bigger than your shelves or drawers and fold the edges under for a much nicer look. It just lays right on the shelf, no adhesives so it's simple to replace. You wouldn't want to get too much water on them but small drops evaporate quickly.

@lindsey ditto! same spelling too! I once had a girl named Lindsay tell me my mom must have been drunk when she named me. What a peach.

Gotta watch out for the Lindsays. Lol