Well, there's you problem. I have traveled the world, including many trips of 6 weeks or more, with only a single. carry-on. Laundromats exist everywhere.
Snark aside, a bit of planning and organization w.r.t. packing pays off handsomely in terms of flexibility when traveling.
Spontaneous weekend trip to a remote island when carrying only a backpack - sure. Add two heavy suitcases, and it becomes an impossibility.