My bet is repair will exceed replacement cost but yes, almost all can be repaired. There is one thing and the battery and charger must be from Sony. Due to battery fires generic batteries may not work.
I have a Sony VPCF136FX. The battery stated it was 100% but would not run the computer. I bought a new battery and it worked with the battery power it had but would not charge. The strange thing is if the battery and AC adapter or plugged in, it thinks its on battery but wont charge. If I take the battery out, it recognizes that its on AC. I have tried deleting the Battery in the Device Manager. Is this a hardware issue and can it be repaired?