The amount of space is going to vary greatly based on your plans. Some people keep goats in relatively small pens without any issues, but that doesn't provide much opportunity for browsing/grazing and requires more hay to be included in their diet. You will want separate areas for the goats and chickens to eat. Chicken feed is bad for goats and can be fatal if enough is consumed. Chickens also have a tendency to soil water buckets and such, and goats do not like dirty water. If goats are kept in smaller pens, you will want to provide them with opportunities to walk/run and get some exercise either in a larger area of on leashes/lead ropes.
you could probably get away with something as small as maybe a 10X10 BUT it’s going to have to be cleaned very often, and the goats are probably not going to be overly thrilled about having such small living quarters. Kinda like rabbits and their tiny little cages. Technically the space is fine but I just don’t see how those are happy rabbits. So what I would do is look at how much space you have that could be “donated” to their pen and just fence it all in. The more space the happier the goat, the happier the goat the healthier they will be I also agree about the chickens. They really can be a pain in the behind. They will jump into feeders and eat the goats grain, lay eggs in their hay and peck the goats so they have to wait until they are done laying and I’m not sure why but they think roosting on feeders is the best place in the world to sleep. There are a few (I’m really not a chicken fan lol) good things about them. They do help with flies and other insects. So depending on your set up and what you want you could have a small pen for them to go in at night, also where their food is so the goats can’t get to it, and let them out during the day. The smallest space for a chicken is a 1 square foot per chicken. I would only go this small if they were only put up at night. I don’t know I have a thing about animals in small spaces.
I use what is called hog or cattle panels cut to size with bolt cutters and attach the sections with clavicles to eye bolts in 2x4s across back walls and 4x4 posts down isle ways. This allows me to change the size of the pens as needed. You should be fine with just a regular size box stall for horses 12x12 or 12x14 for a couple of goats. They are nest builders and the smaller pens are harder to clean until you learn to start at the back with a curved hoe and roll the matted wasted hay and bedding forward. You have to have room to maneuver your forks and tools. Without banging into walls and not being able to turn. Also you should consider how your going to move stuff outside. Wheelbarrow, ATV with cart, or sled. You need to be able to have that loaded and still be able to get on the other side to pull it out. A lot of thought should go into fencing and the best predator proof method used. They should have an area to be turned out into to graze and play during daylight hours of Ok weather. Goats don't like getting wet so they should have something to get under or be able to get back inside if it starts to rain.
I would build a Coop or Chicken Tractor for your chickens and keep them separate from the goats. Right now is the chick sale days at most farm stores so a good time to get them. I used plastic fencing on my Chicken Tractor and it has worked great for many years.