The nearest beach from the house is that of los Alcazares in Mar Menor and if you want to go towards the Mediterranean Sea, you can go to San Pedro in the north of the lagoon and in the south to Cabo de Palos.
In the south there are some beautiful natural beaches with mountain views. It is difficult to get there but it's definitely worth it. Take the (fast) road towards La Manga / Cabo de Palos (RM12) and take exit number 10. Turn right onto the long straight dirt road, keep right after two kilometers and follow the signs "playa". The beaches of Calblanque are empty and pristine. Expect no beach bars.