Executing the code
r = RegionIntersection[Region[Ball[3]], Region[Ball[{1/2, 1/3, 5/6}, 1]]];
Region[r]
in version 11.3.0.0 on Windows 10 32-bit, I don't obtain any plot, only unclear output with "Embedding dimension: 3". However, this code works on Wolfram Cloud (see screen ) and on 64- bit Windows (as I was kindly informed by Technical Support [CASE:4195740] ).
Addition. In view of the above I can execute
Volume[r]
on neither my comp nor the cloud.
Addition 2. The reinstallation of Mathematica on my comp solved the problem. I am grateful to Duncan Pettengill for his well-meaning and professional attitude to my case.