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There’s a small airfield outside Siena, but it’s a military airfield – not a commercial airport.
The closest airports to Siena that you might consider are in Florence or Pisa – but it would take far more time to fly from Rome to Florence or Pisa and then take a train or bus to Siena from there than it would to simply take a train or bus directly from Rome to Siena.
In other words, don’t try to fly between Rome and Siena. It’s just not worth the trouble.
photos, top to bottom, by: Stanca appena sveglia, moty66.ipernity.com, miquitos, Rosa Menkman