What remains comparable is how much humans need to eat in a day. So it could be meaningful to work out how many people a day's wage could feed back then and use that as a baseline to calculate the imaginary USD/DRI exchange rate.
Think of it as "scarcity distribution" if you will. The typical food basket cost a different amount then than now. And there are plenty of times when most people could barely afford to eat (e.g. the early industrial age, when most factory workers were barely paid a living wage).
A good example of the differences that make this discrepancy so large is the price of salt. Just now, the department in charge of road gritting in the UK has announced that it's out of gritting salt, so they're planning to use table salt instead. Flash back two thousand years ago, and salt was so precious that it was used as a currency to pay soldiers (hence the name "salary").
Other commodities suffer from similar differences, not only with 2000 years of distance, but even closer to us. As pg pointed out, many things which are available today for a dollar were unthinkable even 50 years ago. All these discrepancies make it impossible to compute any sort of exchange rate between dollars and denarii. You can make one up in your head, but it's pure fantasy.
I don't deny the difficulties at all and these difficulties are hotly debated among economists even today when comparing say China or Cuba with the US. There is no completely satisfactory solution, but concepts like purchasing power parity are not completely meaningless even when different types of goods are consumed.
I agree that it gets very vague when societies are as different as ancient Rome and a modern industrial country. But knowing what share of a person's income is spent on biological necessities (basic food and shelter) is meaningful and can be expressed in dollar terms to make it more understandable.
