You're selling the June contract right now. There will be nothing to unload. Whoever is holding your June contract when it expires will be the one left holding

You're selling the June contract right now. There will be nothing to unload. Whoever is holding your June contract when it expires will be the one left holding the bag (well, the barrels). That's the whole point of futures. The problem is that you have to store the oil for a month which is pretty expensive right now.