Anyone Used Samsung Gear 360 Camera Inside Vatican or Colosseum?

Here is the policy on photography in the Vatican Museums. It looks like cameras are not restricted by size, but you cannot use a flash and photos with "electronic equipment" are forbidden in the Sistine Chapel.
"It is permitted to take photographs, for personal use only, of the works in all areas of the Vatican Museums, apart from the Sistine Chapel. Flash photography is strictly forbidden.
The use of tripods, stands and/or professional equipment is not allowed; any eventual use must be authorised by the Directorate of the Vatican Museums.
The use of telescopic "selfie sticks" is forbidden.
In the Sistine Chapel it is forbidden to take photographs or films with any type of electronic equipment. The guard staff is authorised to request the immediate cancellation, in their presence, of video or photographic material produced in contravention of this rule."

http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/consigli-utili.html

Elsewhere in the Vatican the policy may vary -- the Vatican is an entire city, you realize.

Here are the policies for the Colosseum; AFAIK it doesn't say anything about the size of cameras allowed.

http://www.coopculture.it/en/the-colosseum.cfm

I don't think there's any problem with the size at either place as long as you're not using any sort of attached flash, monopod or tripod. To be honest, I think your bigger issue is going to be shooting 360 around the crowds at the Vatican museums.

As mentioned, no photography of any sort allowed in the Sistine. To disagree slightly with the previous poster, while the Vatican is called a "city", it's more of a walled complex, not really very big, and much of it is off-limits to visitors. You may see more of it than the average Joe depending on how much of it you intend to book tours for: you may only see the museums, basilica and St Peter's square on your own. The gardens, Necropolis of Via Triumphalis and tomb of St. Peter scavi/necropolis (no photography allowed) may only be accessed with a guided tour.

From my experience, you'll be hard-pressed to take any worthwhile photos, must less 360 photos, in any tourist location in Rome. There are just too many people (unless you take an absolutely private tour). The Vatican Museum tour tends to be shoulder-shoulder and lock-step for the entire length of the corridor. Additionally, in the name of antiquity conservancy, the lighting isn't very bright so good photos may require a flash which isn't allowed. On top of all that, if you're on an escorted tour, you need to contend with the pace of the tour guide...

The Colosseum also tends to be really crowded...

Also to be aware of, European Union privacy laws are pretty strict. Just go to google maps street view and see how all faces and license plates are blurred-out. While in Italy two years ago, I witnessed tourists (or possibly non-tourists), on more than one occasion, complaining rather bitterly about having their pictures taken without their explicate permission. On one occasion, a security official made the picture-taker delete the photos in question.