I don't think so, and I don't think they should share that information, but they've given some "clues" (potentially false) over the years with references to things like "over 1 million users." (Though "user" is a vague term considering how their system works.)
In response to a similar question in 2006, Jason Fried said:
We’re a private company so we don’t disclose this information.
My model was largely based on a few key numbers they had published and a general freemium business model. Since then, they've dramatically reduced the prominence of their "free" option and also stopped posting any details of their business, so I haven't been able to realistically update my analysis.
That said, I think that by focusing much more on their paid plans, and by branching out into even more business lines has made 37signals more and more profitable. (See their recent string of new hires.) I wouldn't be surprised if they were exceeding $10million a year in profit, which would be ~$5million a year to DHH. Enough to safely afford a dream car like this. :)