This turned out to be a MIL-DTL-5015 part (Formerly MIL-C-5015), not a MIL 38999 part. It is also known as "type 97".
There is a Digikey catalog page for this part, and they even have 34 in stock. The catalog page for it has a nice image of the pinout:
https://www.digikey.com/product-detail/en/amphenol-industrial-operations/97-3106A-28-21P/97-3106A-28-21P-ND/341000
Here it is at Newark:
http://www.newark.com/amphenol/97-3106a-28-21p/circular-connector-threaded-circular/dp/09WX9540?MER=sy-me-pd-mi-alte
The way that the part number works is that the last two numbers, 28-21 are the insert type. For some reason neither of these numbers match the number of pins but that's how things are often done by military contractors. Here's a datasheet showing the insert options:
http://www.farnell.com/datasheets/2158844.pdf?_ga=2.223425371.1359845224.1503779982-1653510471.1485545260
Glenair has a lot of MIL-DTL-5015 hermetic connectors, and feedthroughs. I don't presently have an answer for what distinguishes their "Type D" MIL-DTL-5015 feedthroughs from their "regular" MIL-DTL-5015 feedthroughs. For some reason, Glenair seems to have multiple pages for these connectors that have the same information.
https://www.glenair.com/hermetic_connectors/g.htm
https://www.glenair.com/interconnects/mildtl5015/
Mario's vacuum chamber uses 37 pin feedthroughs by Douglas Electrical Components with a part number on the side of SET28896. Sadly, this part number can not be found anywhere on their website or in their current catalog. The catalog does make references to MIL-C-5015 feedthroughs, with one option being 37 contacts, but the table does not seem to have information on how to order or availability.
http://www.douglaselectrical.com/brochures/dec_product_catalog.pdf
Interestingly, Pave Technology does seem to sell these feedthroughs, although their site doesn't offer the option to download drawing PDFs if I'm using IE:
http://www.pavetechnologyco.com/design/pm1_productindex.html
