The second home is not in the United States. My great grandfather bought it in the 1940's. The government there used Eminent Domain to force out everyone, including me, in the apartment complex. It paid me compensation last year in 2019, which was then converted to US dollars. I do not know the purchase price of 80 years ago as I do not have any old purchase documents. The government records office there said there was no purchase price recorded. Do I have to report cost basis as zero, which means I have to pay huge long-term capital gain?
You said the apartment was inherited, but it's not clear whether you inherited it directly from your great-grandfather, or there were intermediate transfers of ownership. How did you become the owner of the apartment? If you inherited it, your basis is the fair market value on the date of death of the person you inherited it from. That should be easier to determine than the price that your great-grandfather paid in the 1940s, and will be considerably higher than the original purchase price. You can probably get a retroactive appraisal from a professional appraiser.
In most cases, your cost basis is the value on the date you inherited it, not the original purchase price. However, this may have to be adjusted if the apartment was rented or used for business after you inherited it.
Be aware that if you are audited, the IRS does not have to credit you with any basis that you can’t prove. So to avoid paying tax on the entire sales price, you should make a diligent effort to determine the fair market value on the day you inherited the property. There must be local appraisers or real estate professionals who can help with this.