In theory, it would be possible to build a house out of cardboard. But cardboard is probably more useful as a source of recycled fibers than as a construction material. Building a house out of cardboard would effectively take all of that cardboard out of the recycling loop in order to displace other materials such as wood and straw that can be produced sustainably but are not as easy to recycle. So I think in most cases it isn't a good sustainable building material.
If you really want to build with the cardboard, for example if it isn't possible to recycle it in your area but you'd like to put it to good use, you'll need to protect it from fire and moisture. You'll probably need to use chemicals for fire protection, but the moisture issue probably just requires some attention to detail when installing vapor and moisture barriers during construction. If the cardboard gets wet, it will weaken and deteriorate, so water protection is very important. It will be nearly impossible to dry the interior of a "solid" cardboard wall if it gets wet. For practical questions about fireproofing and moisture protection, I recommend paying DIY Stack Exchange a quick visit. As far as cardboard's ability to support the weight of a house, I don't think there will be a problem--cardboard is an impressively sturdy material--but it might compress a little bit over time. A carefully built cardboard house could be durable and efficient, but you will probably need to get help from construction experts to achieve that goal.
A cardboard house is possible, but without clever engineering, it's probably not worth building. I strongly encourage you to engage in clever engineering. A durable, efficient cardboard house that beats wood, masonry, concrete, and straw bale in cost, efficient use of materials, and performance would be a truly wonderful invention. But I don't think a straw bale-style cardboard house will do that.