Maybe a small mouse?

My strawberry problem is squirrels!

If it is the bottom half that disappears, I would check the soil for bugs or cutworms.

It can be hard to figure out what is eating one's strawberries merely because there are so many creatures who like them. We have most of the berry-stealing creatures fenced out of our garden, so when something gets into the garden and eats them, it usually is something that has circumvented the 8' tall fence---normally squirrels or birds. Occasionally it is a turtle who squeezes under a specific spot along the fenceline. I haven't blocked his secret spot because he likes to eat Colorado Potato Beetles and other bugs, so he is beneficial in general in the garden. Occasionally it is pill bugs or sow bugs. To control them, I occasionally sprinkle an organic slug/snail bail called Slug-Go Plus, which is a combination product containing iron phosphate and spinosad. This same product also controls cutworms if you end up finding them in the soil around the plants.

Some small birds even find ways to get through bird netting.

Adult rabbits can stand on their hind legs and reach pretty high, so there's still a slim chance they're your culprit. I've had them stand on their hind legs and reach up into molasses feed tubs and eat cherry tomatoes that were hanging low on the plant, and molasses feed tubs are fairly tall.

Good luck figuring out what it is and then how to exclude it. I have more trouble with critters eating strawberries than everything else in the garden combined.