Yes, they can definitely be housed indoors, in a cage or not. If kept in a cage, it should be large enough for them to move around comfortably. Many people use old dog kennels (you can put a shelf in it to give more floor space), or you can use NIC cubes and build them to practically any size, or you can use an exercise pen for pets, and of course there are commercial cages for sale, just be aware many of them are much too small. In addition to the cage space, they should be given a chance to exercise daily, which means their exercise area still should be bunny-proofed for their safety. The biggest thing is electrical cords and house plants they can chew and harm themselves (and of course they can be destructive to your home as well - baseboards, carpet, furniture, etc).
i keep my foster rabbit in my home in a very large NIC cube condo (4 stories) with an attached 4' x 6' run that she has access to 24 hours a day. i wish i could give her more exercise opportunity, but i cannot because of my other pets and my husband's allergies. She sometimes gets to run around in the room where her cage is, but it's not daily, and she's a bit overweight as a result, despite me cutting out her pellets and just feeding her a hay/greens diet.
i line the litterbox with a mixture of a paper pellet litter and a softer paper-based litter that has lavender buds added to it, so it smells nice (with hay, of course!). i clean it out every day, and i vacuum the room where the rabbit cage is, and there is no smell. There can be mess, however, especially from little bits of hay and litter that stick to the fur and get left on the floor.