You don’t necessarily need an actual stable, but it is very useful to have somewhere to shut them in if you have to, however it is ABSOLUTELY ESSENTIAL that donkeys have shelter from the rain. This is one of the areas where donkeys are so different from ponies. Ponies have a considerable amount of oil in their coats and can therefore withstand getting wet to varying degrees (my Mini Shetland ponies stand out in the snow and pouring rain, yet their inner coat is as warm and dry as anything). Donkeys have little or no oil so the wet soaks through and they get very, very wet – and if it's cold too, they can easily suffer from pneumonia. This is a particular danger for foals and elderly donkeys. Most donkeys have the sense to get out of the wet, if given a chance, but babies and stallions tend not to (babies haven’t yet learnt, and stallions usually have their minds on other things!).
A well-fitting donkey rug will keep a donkey dry, or an open field shelter works well, or a stable with the door left open. A means of shutting them in (e.g. if they need treatment) is worthwhile investing in – a door or perhaps a hurdle. Two donkeys will happily share a regular sized stable and would probably prefer to share rather than be stabled separately. Donkeys are also likely to use the dark stable to escape from summer midges if they become annoying. Remember what I said about many of them liking wood? – you might want to protect any particularly tasty bits before they literally eat themselves out of house and home!
Any shelter will need some form of flooring and possibly bedding – the flooring / bedding combination needs to ensure that the donkeys have a safe footing, will not slip and that they can lie down free of their own waste (so urine must be able to drain away or be soaked up).
I have concrete floors, with a slight slope on them, and have rubber matting on top. This is quite expensive to set-up, but once there, means little or no additional bedding and minimises mucking out time as there is only the actual ‘poo’ to shovel with no associated straw or shavings. My muck heap is minimal and what is there rots down very quickly.
Straw is used by many and has the added benefit of being an additional food source for donkeys who can pick at it and get an extra source of fibre. Wood shavings are also very popular and very clean to use, and hemp stems (sold as Aubiose) are also widely used where available. Some people also use shredded paper. It’s all a matter of what you prefer, what you can find in your area, and how the handling and disposal of it suits you and your donks.