I also thought this is an almost impossible problem to solve, until a friend pointed out that on the Jubilee line extension (London Underground), it is solved.
The edge of the platform at the newer stations is walled off from the track by glass. The platform doors line up with the train's, and open/close at the same time. So there's no access to the track at any point, hence no opportunity for "person under train".
Platform doors are certainly possible above ground too.
Usually in Tokyo, they use chest-height barriers/doors for above-ground lines, e.g. currently the Meguro-line has these, and in the future, the Yamanote-line will. This doesn't make suicides impossible, but makes them more difficult to a degree which will certainly discourage many people (you can't just fling yourself off the platform at the last moment). [Some subways, e.g. the Fukutoshin-line, use these as well.]
A more problematic issue is stations which handle many disparate types of trains on the same platforms, which makes it much more difficult to achieve the consistent door positioning you need for platform doors.
They do end up saving some money too, though, not only by reducing suicides and accidents, but because platform doors seem to be considered a prerequisite for "one-man operation" (ワンマン運転) on most lines, and that can help reduce staffing costs.
Most Tokyo stations also have high enough ridership that such capital improvements are a lot easier to swallow than they would be for "concrete slab in the countryside" stations.
Luckily, it's the sort of thing they can do incrementally in the places where it makes the most sense; no need to do every line and every station immediately. I imagine they'll probably continue to do it slowly, just as they're slowly adding elevators and other accessibility improvements to even minor stations.