Yes, Palazzo Datini is open to the public!
Ciao scrittore,
I found some information on the Palazzo Datini, which is where the rich Merchant of Prato Francesco Datini lived, but nothing clear about visiting it (it houses State Archives now), so I contacted the official tourism board of Prato to request further information. This is what I got:
The Palace is open to the public:
from 9:00 a.m. to 12:30 pm and from 3:00 p.m to 6:00 p.m , Monday through Friday
from 9:00 a.m. to 12:30 pm on Saturday.
Closed on Sunday and festivities.
On August the palace is open only in the mornings.
Entrance to Palazzo Datini is free.
Hope this information is useful!