Central yet peaceful, cats,old town

My flat is right in the old town and is in a beautiful plaça - near the centre, the beach, a park, train stations (going both north and south), and loads of lovely restaurants and bars. I am studying and working from home some days, so a peaceful visitor would be lovely:) but I am right in the centre so you can easily go out and about on adventures! I have two gorgeous cats so you need to be a cat lover and not have allergies. The flat is not quite wheelchair accessible (the slightly 'amusing' mistake of having one big step before getting to the stairlift that takes you to the lift!), and inside the flat a wheelchair would struggle but wobbly people (like myself) would be fine getting down the corridor. A peaceful and/or adventure packed time in Barcelona in a pretty, quirky, sunshine and cat-filled flat, is what I can offer you:)