There’s no problem in finding ways to get rid of trash. There are plenty of ways to do it, and in fact even just landfilling works just fine. When you hear about a garbage barge wandering the seas without a destination, or whatever, that’s just because of political disagreements about where to put it, not because there isn’t any place to put it.
What’s tricky is finding a way to get rid of it while also turning a profit. Ideally, you’d find someone who wants it enough that they’re willing to pay more than the cost of collecting it. For scrap metal, this is often possible. But for most other materials, it’s usually not, so you have to charge the people throwing the stuff out for the service of picking it up. Even there, you hope to at least get something for the stuff, so you can discount the pickup charge, but even that isn’t always possible.