A Wunderkammer or Chamber of Curiosities, is a 16th century tradition of idealists, who collected at that time uncharted specimen of animals, plants and exotic art. Most content of historical Wunderkammern can today be seen in museums of natural history.
Nowadays, if someone owns one privately, it is usually a collection of arts, fossils and travel souvenirs. I have written before about my love for private libraries and greenhouses and just because megalomania is fun: a Wunderkammer is also on my “want” list.
What one collects is pretty much up to one’s liking; the only “rule” may be that it may be various. There are (modern) Wunderkammern containing corals, stones, taxidermy objects, art, historical instruments/laboratory objects, items in form aldehyde, fossils of all kinds, crystals, textiles, pottery, and even historical job specific items (doctor supplies, historically relevant farming tools etc.). Collections of historical books and maps may be counted as well.
Wunderkammern can be grand museums-like displays or small, cupboard-sized projects. I will show you a few real-life examples from opulent to suitcase-friendly.
If you are rich/have lots of space/a lot of time:

A cabinet displaying everything from butterflies to globes. An extensive collection like this requires a lot of patience and a lot of cash.

A lot of taxidermy going on here. Again a display that requires money and space.

This one is a little more reasonable.
It still uses the whole room, but it requires less dedication to make or upkeep. It shows a few taxidermy objects, a few fossils and some historical instruments and old books. (The floor is beautiful too!)
The classic cupboard option.
Even in the 17th century, it was popular to have a Wunderkammerschrank, Schrank meaning “cupboard” in German.
This option still requires a spacy, sturdy cupboard and a reasonably sized collection, but it is definitely a more realistic option for most.

Bones, corals and nic nacs. A colourful array of whatever the owner found curious. Adding historical images and using a wooden cupboard gives it some class.

Here we have taxidermy, skeletons and corals. This one is great for those who want to emphasize the decorative element of Wunderkammern.
Simplicity and size.
If all this is not reasonable due to cost, time and space, one can always focus on small sized options.

Natural items and objects found in the ocean. A very simple and pretty display with minimal cost.

A simple coral display. Highly decorative and again with minimal costs involved.

And the probably most space-saving option: a wall display.
Now before you go and purchase that antique ceiling high glass case, consider a few things:
Corals, fossils and taxidermy items are generally not allowed to take on travels. If you purchase an item over the internet, make sure it comes from a location that is legally allowed to sell the item. If the animal or plant specimen is on the verge of extinction, it shouldn’t be sold at all.
Preserving in formaldehyde may cause risks and is potentially illegal. The European Union has banned the use of formaldehyde for preserving and embalming as of 2009.
Preserved body parts are not permitted to be obtained, that includes foetuses. (Skulls may be obtained but are subject to regulations).
Artwork must be authentic. If a piece of art you obtain happens to be a yet unknown work of a famous and high-priced artist, the reasonable thing to do is to give it to an art gallery.
If you are offered a perfectly preserved fossil, consider two things:
A) The fossil may be a fake.
B) … and that is the more important one – if the fossil is really impressive, it should be placed in a museum. A stand-out object deserves to be displayed and used for scientific and educational purposes.
C) Some people collect “mythical” creatures on purpose, but if that’s not your thing, be aware that offers of unicorns and mermaids etc. are obviously all fake and have little scientific value (of course if you like them, go ahead!).
D) Religious items, specifically bones of “saints”, may belong to a church or an item stolen from churches. A lot of them are also subject to travel restrictions. (And most likely fake).
E) As more value your collection has (especially when it contains artwork, pottery or jewellery), as more carefully you may be about telling. You don’t want to attract “grave diggers”.
F) If your collection is valuable to you or generally valuable, consider donating it after your death. If it doesn’t contain contemporarily valuable items but has great decorative and personal value, bequeath it to your family. If it does contain valuable items, consider bequeathing it to a suitable museum or university.
Now, have fun!