the battery in my old white lap exploded and cannot be turned on. Can this be fixed?

Your best option is to take your laptop into Apple Store for an estimate. Even if your Apple Care has ended, some problems like an exploding battery still might be covered.

Otherwise you might be able to remove the drive and insert into an external drive for recovery.

http://www.wikihow.com/Recover-Data-from-the-Hard-Drive-of-a-Dead-Laptop

Or get new battery for your MacBook and install it by yourself, however as dianeoforegon told, you'd better take it to genius bar.

I think your MacBook works fine if you replace exploded battery.

http://www.everymac.com/systems/apple/macbook/index-macbook.html

How to identify MacBook models

Some exploded batteries may have initially expanded and damaged the housing and the circuitboard in the process. So your mileage may vary on that one.