It will depend. A couple of years ago I went to an onsen with a friend who's heavily tattooed across the shoulders and legs. We explicitly asked at the reception desk whether that's okay, and they said it's no problem. After we were in the onsen for some 20 minutes or so however, an employee approached him apologetically and asked him to leave after all. From what I could hear between the lines, other patrons had lodged a complaint, and they bowed to it.
Fast forward to a few weeks ago and I met a (very nice) middle aged Japanese guy at a local sento who was so heavily tattooed across his entire back, that he was surely a "yassan". But nobody minded.
So, if you don't obviously have tattoos, it'll depend on how that's perceived. If you can explain it away as accidental scars, surely nobody will mind; you might get some pitying looks however. If it's obviously some self-mutilating body art kind of thing… you may get asked to leave if it freaks others out too much.
You may want to look for onsen which do not explicitly forbid tattoos or even explicitly allow them, then your chances are probably better. If you can somewhat cover up your scars with your washcloth, it may be a good idea to do so. The less you flaunt them, the better your chances.