Here's the problem: you have the burden of proof for everything you claim on your tax return. You might be able to find a 35 year old stock price from the company's web site or newspaper archives in a library, and you could prepare a tax return using your best guess of what your cost was. But if audited, the IRS does not have to give you anything you can't prove. The IRS could assess capital gains tax as if your cost basis was zero.
Do you have any records at all that would at least show you have owned this stock for more than 1 year? If you can't even prove that, the IRS could deny you the long term capital gain rate too.
