Select 4 out of 12 items?

I am sure that it is possible – but you may need to be creative.
Using Product Options you can set up a rule which has 12 radio boxes (let you choose multiple options)
linked to 12 product options each with a different price variation.
A customer can click on 4 check boxes and those 4 product variations will be sent to the cart. That assumes that they are all variations of the 1 product.
That may limit the amount of information you can have on each variation.
If you want to handle them as 12 different products,part of a group set maybe then you could set them all up under the same Category , so they will be displayed together on the same page and same menu, then maybe you could get a short description and a radio box displayed next to each one in Category view.
It may be much easier to control if you build your own page using shortcodes,
look at https://default.readyshoppingcart.com/insert-product-in-to-the-post/
The tricky part would be limiting the selection to 4 , so you would need to be able to write some code for this.

The problem got a little bigger.

The client has a line of 21 products they want to put in baskets – it’s a mix and match sort of thing.

So on one page the customer needs to be able to click on the time, then enter how many they want in the basket and this needs to be done several times.

Let’s say the items are oranges $1, apples $2, bananas $3, grapes $4, pineapples $5, strawberries $6.
The customer needs to choose which out of those items they want in the basket and get a total.

So if the customer wants 2 oranges = $2, and 3 apples = $6 ($8 total)
Anyway to do that all on one page?