280k homeless?!?!? Japan claims they are down to ~4k nationwide and AFAIK they don't have nearly as much support for the homeless (could be totally wrong on that). What they do have AFAICT is a culture of self reliance so that someone who is homeless generally feels guilty about asking for help (not saying that's good or bad thing, only saying that's what I believe is the culture)
They aren't often easy to see but if go into a public park, if you see blue vinyl tarps in the bushes those are often a homeless person living there. There's a few in Yoyogi park and some in Shinjuku Central park (next to Tokyo's gov HQ). Otherwise in the winter some people try to sleep in the subway stations close to closing time.
Japan does not have "free" medical support AFAIK. You have to pay into the national health insurance or otherwise pay your own medical bills. On the other hand, prices are regulated.