Suddenly both keys are not working correctly. Batteries in keys are good. Doors will open/close only if pressing the key buttons near front windows. Car will start only if the keys are near the central console or in it's place. What might cause this?

 

A bad antenna? (had that problem with my 2015 xc70, but had a tire pressure system service message also.)

 

Which antenna was it? This seems to have 7 antennas around the car. Maybe the black box in the trunk is faulty in some way?

 

That antenna is in the very center of the car between the headliner and the roof. You have to drop the headliner from the rear slightly to get to it. In 2015 here in the US we have direct tire pressure sensors in the xc70. A sensor in each wheel reporting low tire pressure versus the indirect system that uses the abs sensors in some markets. My first problem was the "tire pressure system service needed" message appeared. Then later the remote start/lock/unlock range reduced to where I ended up having to use the emergency key to unlock the car. Sourced a used antenna (because I was not sure that was the solution) and problem solved. That antenna is used in many years of Volvos - my replacement came from a 2013 xc60. Note the different remote and receiver frequencies for different markets.