Unless they said otherwise–or art gallery positions (about which I admit I know next to nothing) are different from most other jobs–I would say the default is not a handwriting sample.
From what you have said, I think the most likely explanation is that writing will be a part of the job, and they would like some way to assess your writing ability.
Ideally you would send them a piece of writing that is identical to the kind of writing you will be doing for them. For example, when I was applying for attorney jobs, I sent a sample of a brief that I wrote.
If–as appears to be true in this case–we’re not completely sure what kind of writing they expect you to do, then we have to make an educated guess.
I would say that the art gallery is a business, and that they want to know that you can handle business communications. Letters to artists and customers, newsletters about upcoming events at the gallery, a memo to the staff about a new policy regarding overnight guests (ok, that one was a joke).
I would say they would be looking for straightforward, expository writing, such as memos, newsletters, papers you wrote for a class (even one that has nothing to do with art), business letters you wrote in a prior job.
Something between 2 to 5 pages that shows that you know how to express your ideas, that you have a command of grammar, that you know how to proofread. Although you are applying for a job in a creative business, I would tend to lean more towards the business-like than to creative writing.
What are the last 5 things you’ve written? Let us know some options and we can help you pick the best one.
Constantine