A fungal infection of the feet is one of the most likely possibilities (athlete's foot). Wearing sandals does not mean you cannot get athlete's foot. Although it is true that sweating and moisture tend to allow athlete's foot to proliferate, not having your feet protected (by wearing sandals) can be a good way to pick up fungus from the soil and other environments. Athlete's foot causes significant dryness, scaling, and peeling of the skin on the soles of the feet, often but not always with itching and an odor. Luckily it is usually easily treated with good skin hygiene and an over the counter antifungal cream.
, which is an inflammatory reaction of the skin characterized by dryness, itching, and peeling or cracking can also occur on the bottoms of the feet. The type of eczema which primarily affects the palms of the hands and soles of the feet is called dyshidrotic eczema. It is typically treated by keeping the skin clean and dry and well moisturized and by applying a steroid cream during flares.
I would suggest talking to your primary care doctor
in order to obtain a diagnosis and decide what the best treatment is.