My son is 6, and experienced this a lot more when he was younger, and sometimes, he still doesn't want to go downstairs or in another room alone, or apart from the family. My 2 older children weren't like that. So, with him, we would walk to the top of the stairs and wait there for him, while he could still see us while he got what he wanted downstairs. Or, he would call "Momma?" to check if I was still there when he went to his room for something and I was in the kitchen. I began to make a counting game of it and distanced myself or remained seated while he went - "I'll count to see how long it takes you to come back, or how fast you can get it." "On your mark, get set, go! 1... 2... 3... 4..."
Eventually, he became more independent, but still calls for me or his dad when he's in another room - "where are you? watcha doin'", and then he'll continue to play. The interesting thing is he never did it when he was outside. He could play outside on the deck or in the yard without a concern in the world, but in the house - he was always concerned with being alone.
Keep steady, try not to play into the fear too much, and ask your pediatrician if this phase is normal and if s/he has any advice.
Pray with your daughter about her fears, if you are a believer. I have a book about a cow who was afraid to go out of the barn to check on a noise at night. The cow woke up another animal, who woke up another animal, and it was a domino effect until all the animals in the barn were awake and afraid of a very inflated noise situation (like the telephone game). The moral of the story is that God is with us even when we are alone. I cannot remember the name of the book. If you are interested, you can email me at home and I will locate the book and provide you with the title if you want to purchase it online somewhere. My email is ____@____.com
You'll be fine.