Crossbow? IMO no, they are basically single shot. If a person only has one home invader they might do the job, but don't miss. On the other hand a bow is an excellent choice for HD, so much so that I do infact keep a 45# bow strung with 6 arrows with broad heads at the ready. When weather is nice I practice daily, and I can hit a paper plate at 15 yards instinctive shooting at a good pace, three arrows in less than 10 seconds. The bow is my primary HD weapon, the arrows are only a danger to the invader, they will not go through several walls like bullets. They also do not cause harm to the hearing, though they do make a noise, but will not be heard outside the house.
The disadvantage is they are cumbersome inside a home, a person needs to practice inside the home or a tight space like a garage to get used to shooting from confines. This means a very low weight bow, and arrows with shock tips to lower chance of damage. A good sized carboard box stuffed with rags, or a good sized piece of plywood using a shock tip designed to only impact not penetrate. These are sold to archery gamers that participate in competitions like paintball. Do not do the in home shooting until shooting outdoors has become proficient, consistent, and accurate. Even a nerf arrow can do property damage if it goes wild.
My bows are PVC, and fiberglass rod bows, DIY. They can be left strung without worry, I have only lost in each about 5# draw from them, after that they maintain their draw weight. They are not fast enough for hunting, but some people use them, I get about 140fps with a 400 grain arrow. That is less than the 200 to 400fps of a compound bow, depending on the cost of the bow.
IMO a crossbow would be even more cumbersome in a home since the limbs outstretch horizontally instead of vertically.
These are the arrows to use for indoors.