I am happiest when I am able to forget myself - there is no pain, nagging thoughts, or acute problems that I have to reflect on or that force themselves into my mind.
Being in motion can help - exercising or just riding a bike or walking outside. Being absorbed in something interesting to look at. (A wood pigeon is sitting on her eggs just outside my kids' bedroom window.)
Making music - being in the flow as Kenny Werner and others have described it. It's not me playing, it's the music manifesting itself through my body. I am not really there, or at least, I am not doing anything. Those are moments of happiness.
I personally have no experience that indicates that happiness is a state that can be achieved - like a wallpaper that is always there and affects the whole character of the room you live in, so to speak.
I think the best we can aim for is to experience moments of happiness - just a couple a day to remind ourselves that it is possible. Maybe this can be practiced and thus gotten better at?