When a bag of holding is destroyed, the contents are scattered in the astral plane. If a person is inside the bag, what happens to their body? According to the accepted answer here: "Will loot you find on the astral plane while using astral projection remain when you leave it?" - you don't have a physical form in the astral plane. Is their body gone, forever? Or do they get a new body when they enter a physical plane?
If a person is inside the bag, what happens to their body?
That person is now physically on the Astral.
Is their body gone, forever?
No it is on the Astral. I think you are conflating the effects of Astral Projection with physical planar travel. The poor individual that was kidnapped and placed in your Bag of Holding still exists physically on the Astral. You could use Astral Projection to find that person but can't really interact with it in a meaningful way. So I would say since your Astral Form is a duplicate of your physical and you don't seem to have a physical form you would not be able to carry any loot out of the Astral in this way although you could reasonably guide the poor individual around the Astral.
Or do they get a new body when they enter a physical plane?
The person in the Bag of Holding keeps the body they have on the Astral plan since they are physically moving through the plane and not projected through magic as with Astral Projection.
As Matt Vincent stated Astral Projection and Plane Shift are ways to get to the Astral plane, however that Wiki is not "official" as far as I can tell and it leaves out the Gate spell as well as Color Pools.
Gate PHB p244
The portal has a front and a back on each plane where it appears. Travel through the portal is possible only by moving through its front. Anything that does so is instantly transported to the other plane, appearing in the unoccupied space nearest to the portal.
DMG p47 for Color Pools
For everything other than Astral Projection you are physically on the Astral Plane, just like any other plane of existence.
Canonically, you can enter the Astral plane physically. For example the FR wiki mentions: "Entering the Astral plane could be accomplished in one of two ways: projecting your astral form into the plane, or by physically entering the plane."
They make it sound terrible, what happens to my character?
As the description indicates, you are "dumped" into the astral plane, with your physical body, still wearing your items. Only Astral Projection allows creating a duplicate body, upon traveling through a color pool to a different plane from the astral. You would have your possessions, and would otherwise be fine.
Note you can quickly find the other items that were in the bag by simply thinking of them. If more time has elapsed before you look for the items, the DM may rule other entities have found them, or it takes a day or so, increasing your chances of an encounter, as the items continue to travel without resistance. Thought is the only resistance, which objects are incapable of.
There are numerous source books describing this (Based on another answer, I see the DMG 5e maintains the gist of previous editions. Most editions are basically the same), but all you need to know is in the following summary:
How Can I get Out?
The Astral Plane is a big place, silvery in color, and every thing moves at the speed of thought. Viewing distance is limited to 600'. Simply thinking of where the character wants to go (probably back to his prime world, but not a requirement) will cause him to move at the speed of thought to the color pool associated with that plane (travel time 6-24 hours depending on intelligence). Note color pools are 99% one-way (Astral to other plane).
DMG, pg 47:
Finding the right color pool is a matter of chance: locating the correct one takes ld4 x 10 hours of travel.
This is where the DM comes in, because the traveler cannot find a particular pool to the world, just a pool to the world. Concentrating for 1d6 minutes allows you to see on the other side of the pool (to avoid coming out in a volcano).
You should be subject to random encounters per day, generally from CR 10-40+. See DMG, page 48 for brief overview. Creating this table give DM's great amusement, so be ready for the worst. Adjust accordingly. For Example:
90-00 Roll on next table
55-59 Astral Dreadnaught
60-65 Astral Winds
66-69 Other mortal(s)
70-75 Githyankie Ship or party
76-79 Githyankie City
80-85 Great Wyrmm Dragon (roll for color)
86-89 Astral Dragon
90-91 Random Fiendish party
92-93 Random Celestial party
94-96 Law/Chaos Creature
97 Body of Dead God
98 Random Good deity
99 Random Neutral deity
00 Evil God
Note the 00 will just devour the character, without rolling initiative. The gods don't generally live here, but may travel the Astral. Traveling the Astral Plane can be dangerous.
Note also Astral Winds can cause confusion lasting 1d6 to 1d10x10 days while being cast adrift. DMG calls it "Psychic Wind," see DMG pg 48 for alternate rules on that
You do not age while on the Astral, nor need food or water. When you return to another plane, the aging comes all at once, however you will not die from starvation or thirst.
DM may rule other entities have advantage to attack in this unfamiliar environment.
You also cannot use Astral Projection to enter a color pool to the originating plane.
- I am not certain but I believe Gary Gygax maintained a similar encounter table for the Astral Plane....can't find my copy.
- Only top level of outer planes can be reached through color pool.
- To get to inner planes you need to go through the Ethereal. So spells like Etherealness, Blink, Conjure Elemental etc. won't work here, nor spells dealing with time/space like Time Stop, Haste, Disintegrate, Sphere of Annihilation, also Maze, Mordenkainen's Magnificent Mansion, Rope Trick. Feeblemind will keep character from moving, as movement is based on Intelligence. Etc.
- Specifics may very edition to edition, but this is the gist.
 Monte Cook. A Guide to the Astral Plane. TSR. 1996.