Once a patient starts on dialysis and is registered with a dialysis center, provided that they are medically eligible they can be placed on the deceased donor transplant list.
Medically eligible being the two most important words in that sentence.

Practically this means that you can’t be 103 years old with a dickey heart, you can’t be an uncontrolled diabetic, you can’t be an alcohol or drug abuser. You can’t be a drug user of any type, and they’d prefer it if you didn’t drink at all. Smoking? Nope! Gotta quit. You can’t have hepatitis or any of another wide variety of chronic infectious diseases that immune suppression would make worse. You can’t have cancer unless you have renal cancer in which case you’ve got to get rid of the cancerous kidney or kidneys. You can’t be morbidly obese. You have to have proved that dialysis will help you. And you have to have the intellectual capacity, time, income, and insurance (thanks free market system) to manage the after care.
In “The Plague and I”, her book about her time at a tuberculosis sanitorium, Betty MacDonald (of Mrs. Piggle Wiggle fame) said, with regards to the intake nurse, “She gave me to understand that tuberculosis was something just a little special and she wasn’t sure she was going to let me have it.”
From a scientific standpoint I absolutely understand the meticulous testing that is required. This whole thing is, duh, a very major procedure and it will change your life. The transplant center wants to be sure that if they go through all the effort of taking a kidney (any organ, really) out of one person and stuffing it into another that the effort will be rewarded. They have to be really sure that they want to let you have it.
From the standpoint of someone who loves someone whose kidneys were failing it looks like an almost insurmountable pile of hurdles that you’ve just been asked to jump over.

One legged.

Physical exam, full blood count and blood chemistry panel. Abdominal ultrasound. Chest x-rays, cardiac ultrasound, cardiac stress test. Sometimes angiography. Colonoscopy. You have to have meetings with social workers, with nephrologists, surgeons, pharmacists, and with transplant center associated financial consultants.
Andrew and I were incredibly lucky that we both work for very understanding organizations. And had we been living anywhere less urban than we do — which is to say if we’d been living anywhere with less access to all of these services — getting to “medically eligible” would have been very difficult indeed.