Keeping dogs as pets which means they live with you inside the house is haram because the Prophet (salallahu alayhi wa sallam) said the angels of mercy do not enter a house in which there is a picture or a dog. (Bukhari) It would be a different matter if you had to temporarily shelter a dog from harm and release it as soon as the danger passed. But taking in dogs as you mentioned means there will always be a dog(s) present which would make their presence in your house constant. This would ultimately lead to a deliberate disregard of the hadith of the Prophet, salallahu alayhi wa sallam.
I truly commend your compassion for Allah's creatures. But you should at alternative ways of helping them. Remember animals in your care would need to fed and provided treatment for if they are sick/injured. It's not cheap. And you could fall into neglect and incur sin upon yourself. Also considering, you're not allowed to keep pets outside, where will the dogs go to poop? You need to be practical about finances, the dogs’ needs, and be able to provide outdoor accommodation to facilitate this project successful on a long term basis. Even so, you should ask a scholar if this is permissible. Animal shelters all over the world have to put down animals because they just simply don’t get rehoused and become impractical and expensive to keep in large numbers. How will yours be any different? Islam also prohibits the sale of dogs. So in the case this venture was permissible, you'd be either paying for it out of your own pocket and/or relying on public donations to get by. I think you're a kind and sensitive human being but one who's overlooking practicalities and letting compassion get the better of them. May Allah be pleased with you.
I heard Allah's Apostle
saying; "Angels (of Mercy) do not enter a house wherein there is a dog or a picture of a living creature (a human being or an animal)."
Hadith - Bukhari 3:515, Narrated Abu Huraira
Allah's Apostle
said, "Whoever keeps a dog, one Qirat of the reward of his good deeds is deducted daily, unless the dog is used for guarding a farm or cattle." Abu Huraira (in another narration) said from the Prophet, "unless it is used for guarding sheep or farms, or for hunting." Narrated Abu Hazim from Abu Huraira: The Prophet
said, "A dog for guarding cattle or for hunting."
[Hadith - Bukhari 3:515, Narrated Abu Huraira]