I’m sorry about the loss of your dog.
Rabbits can make wonderful pets, but they can also be a lot of work, and need specific care that is different from dogs. It is very wise to do your research and make sure that a rabbit is the right companion animal for you. A lot of people don’t realize their care and their ways, and aren’t prepared to care for them for their lifespan, which can be 10 year or more. There is some very good rabbit info on this site. Go to this link:
https://binkybunny.com/BUNNYINFO/tabid/53/Default.aspx. It covers all the basic care aspects.
Rabbit are clean animals and do not smell. They can be litter box trained, although some rabbits are better at that than others. It would of course smell if you did not clean out their litter box on a regular basis. It’s best to remove the dirty/wet litter every day. You would need to make sure that you use a litter that is suitable for rabbits, many of them are not. It is easier to litter train a rabbit that has been spayed or neutered. You might not have much success otherwise.
Rabbits do love the company of their own kind. They love to snuggle and groom each other, but many people do just have one rabbit. If there are people around a lot of the time to keep it company, the rabbit should be happy with that.
Most rabbits are bad chewers, so any room that they are in has to be “bunny proofed”. They especially love to chew on electrical wires. They will also chew on furniture, carpet, wallpaper, baseboards, books, etc. You have to protect your possessions, as well as keeping the bunny safe. Some rabbits can live free range in a room, maybe only being caged or put into an x-pen at night or when no one is there, but other rabbits may always need supervision and not be able to be loose in the house unless they are being watched. Whatever cage or pen you get for them needs to be an adequate size. Bunny proofing and housing are two of the topics discussed in the Bunny Info link above.
The best way to get a rabbit is from a rescue or shelter or humane society. Most of those places will spay or neuter the rabbit, which will save you money, since that is expensive when performed by a private vet. When you do need a vet, you will need to find one who is experienced with exotics. They aren’t always easy to find, but there are links here to help find good rabbit vets.
Read all that you can here and other websites, such as http://www.rabbit.org, which is the House Rabbit Society website, to make sure that now is the right time to bring home a rabbit or other pet. Since you are currently ill, you might not feel up to the work involved with having a pet.
I hope that you feel better soon.