Can I design a photovoltaic system, or is that unrealistic?
It depends what level of 'design' you want to go for.

If you want to design the circuits of the inverters, then it would be an interesting and long learning exercise for you, and the result, in a year or two's time, would be more expensive and almost certainly inferior to anything you could buy.

If you want to choose panels, inverters etc from a catalogue, install them and wire them together, then you may achieve a cost saving over having professionals do it, but not a time saving. Make sure you understand the specifications of what you want to buy very thoroughly. 

Although your completed system could then run your house, you would be unlikely to be able to sell any surplus power to the grid. Your grid connection supplier would only agree to pay you for power if the equipment was to a certain code, and the metering installation had been done by approved vendors. With most people, the principal payback of a home solar system is sale of surplus power, not replacement of imported power, so you'd be missing out out on a sizeable chunk of any revenue if you went DIY. You might find an agreeable and qualified solar installer who is happy to let you do some of the installation yourself, but you can't expect too much discount, as their workload and risk increases through having to verify your work.