Just under two hours by ferry from Dubrovnik is the island of Mljet. At its Western tip, a small national park is very inviting for biking, hiking and swimming.
In contrast to the Old Town of Dubrovnik, the island Mljet seems to be more popular with local tourists – especially for (sun)bathing. On the island, there are two (salt water) lakes which are suitable for swimming: Veliko Jezero (Big Lake) and Malo Jezero (Small Lake), respectively. The latter also sports a small beach which sees quite a few guests on Sundays.
I myself started with a bit of sightseeing, though, and visited the Benedictine Monastery on Saint Mary's Islet first. Afterwards, I enjoyed the cool (and exceptionally clear) waters of Veliko Jezero.
All in all, it was a very relaxing day on Mljet island.