Your acceptance and rates will depend on what type of cancer and how long you have been in remission. I had thyroid cancer, which is a fairly curable cancer, had treatment nearly 10 years ago, and have not had any recurrances. I got private insurance with no increased rates thru BCBS, however they did require a statement from my oncologist saying that my cancer has been in remission for greater than 5 years. They did, however, add an exclusion for any thyroid cancer related treatment, which sucked. I ended up just getting insurance thru my husband, and now on COBRA for as long as we can possibly stay on it!
Can you get a job for the next few months before starting med school to get benefits, then when you quit, get on COBRA. That is expensive, but has group benefits, so you wouldn't have an exclusion for any cancer treatment.