Cala Castrel ? Cala Estremer
 

How do you get to these two beaches, and how long would it take ?

I found this in "Digame" about Cala Castell :-

This wild and remote beach is not for the faint hearted. Cala Castell is 12 kilometres from Pollença. The beach was given its by the 13th century Castell del Rei, which used to defend the coastal area and has breathtaking panoramic views over the north of Mallorca and across to the island of Menorca on a good day.
The isolated beach is located at the end of a cliff and is divided into two coves, formed by pebbles, gravel and some sand. Getting to this beach is not easy. You can take the car to Cala Barques and then from there you will have to start to walk along the coast crossing the Pas des Pescadors, which leads to this privileged natural environment.
There is another possibility of getting here and that is to drive the car until you reach the possessió (rural finca) of Ternelles and then you will need to walk along the track that leads to Cala Castell which is a good 10 kilometres walk. Probably not best to do this in the high of summer. It does however mean that you get the beach to yourself.

getting to Cala Castell
 

To get to the beach from Ternelles you have to get a permit from the council. There are a few posts about this & some useful links. Only 20 people allowed to walk in this area daily.

We did the walk last May and although it wasn't 'beach' weather ie. cool & very windy! It was a great walk to do & well worth the effort of obtaining the permits. Timewise about 5hrs round trip from Ternelles gates. Paths easy, though last part quite tough on tired legs on the way back- a long uphill drag. Castle views stunning. A very peaceful day's walk.

Cala Castell can be seen from the castle area at the end of the walk to Castell del Rey.
It could be seen as a bonus to walk down there and back after visiting the Castilo - rather than the other way round - though it does entail an additional distance to get there. The Castell del Rey is a great walk from Pollenca town but its not just a stroll! (But Brisbane may think it is!) Make sure you take food and drink - there are NO shops or cafes. Also make sure the walk is 'open' if you are going to try it. There are many occasions when it isn't. You will really appreciate a beer or two back in Pollenca on your return.

The beach is very similar to Boquer beach, lots of debris not really a sun bathing beach. I thought it was private beach owned by the banking family March?

Sounds really interesting walk, I'll have to get into some training before the summer. Thanks for the replies guys!!:):)